Use some of the melted butter to grease the interior of a 5- or 6-quart mixing bowl. Set that bowl aside. Pour the warm milk into a mixing bowl and add ½ cup of the warm water, 3 tablespoons of melted butter, the shortening, salt, 2 tablespoons of sugar, the egg, and the yolk. Mix well. Dissolve the yeast in the remaining ¼ cup warm water with the pinch of sugar and allow it to become visibly active, about 3 minutes. Add the dissolved yeast mixture to the mixing bowl and stir in enough flour, cup by cup, to create a somewhat stiff, shaggy mass that’s no longer easily stirred.
Use a rubber spatula to scrape the mass onto a floured surface, and knead it until you’ve created a dough that’s smooth and elastic, adding only as much additional flour as is needed to keep the dough from sticking. Place the dough in the greased bowl and brush the top with more melted butter. Cover the bowl with greased plastic wrap and a clean kitchen towel. Let the dough rise in a warm draft-free spot for 2 hours. Uncover the dough and punch it down with several swift swats with the back of your hand. Turn the dough over in the bowl and knead it gently to redistribute the yeast. Recover the bowl and let the dough rise again until light and billowy, about 1 hour.
Line two shallow baking sheets with parchment paper and sprinkle the paper with cornmeal. If using one oven to accommodate both baking sheets at one time, place the racks in the upper-and lower-third shelf positions. If using a double oven, have the racks on the center shelves. Preheat the oven to 375°F. If planning to apply a topping to the buns, make your egg white glaze and strain it into another bowl so it’s easier to apply. Place your seeds and/or reconstituted onions in small bowls, if using. After the second rise, uncover the dough and punch it down to deflate it. Turn the dough out onto a lightly floured surface and knead it briefly and gently.
To shape burger buns, use your pastry scraper to divide the dough into 10 equal pieces and cover them while working with one piece at time. Shape each piece into an irregular round, then pull up the sides, repeatedly, pinching the gathered ends on top to form a small knot. Continue pulling and pinching until the dough is round and taut with a slightly pinched area on top, flouring your fingertips as necessary. Turn the ball of dough on its side and drive the blade of your pastry scraper down through the waist of the round (not through the pinched end), cutting the ball in half. Lay both halves cut side up, gently opening them so they lie flat. Sandwich both cut sides together, then pinch the cut edges together, sealing them shut. Place the squatty round down smoothest side up, and flatten it quite firmly, using your fingers to shove the pinched seam gently under the bun all the way around. When a bit flatter than you think appropriate, lay the bun (smoothest side up) on one of the prepared baking sheets and cover it with a clean kitchen towel while you shape the rest. Let the buns rise for 20 minutes, starting after the last bun has been shaped.
To shape hot dog buns, divide the dough into 12 equal pieces and cover them while working with one piece at a time. For each bun, roll a piece of dough into an 8-inch strand and flatten the strand using either a straight rolling pin or the heel of your hand. Roll both long sides of the dough in toward the center, and when the two sides meet in the middle, pinch them together to seal shut. Working with one end at a time, lift the dough at one short end and fold it over, rounding off that end and attaching that portion of dough to the seam, going about 2 inches down the length of the seam. Do this on the opposite end and pinch the dough, as needed, to correct the shape. When done, the seam should be straight and the bun about 7 inches long. Place the baking sheet horizontally in front of you. Invert the bun onto the prepared baking sheet, seam side down, close to one end of the sheet. Repeat the shaping procedure with the remaining dough, placing them in a vertical line, only ½ inch apart. (They should merge in the oven.) Tear off two large sheets of aluminum foil and scrunch each one into the shape of the buns. Spray them with vegetable spray and tuck these foil buns at either end of the lineup of buns, giving these last buns boundaries. If not applying a topping, cover the buns with a clean kitchen towel and let them rise for 20 minutes, starting after the last bun has been shaped.
If topping hamburger or hot dog buns, as soon as you’ve finished shaping them, brush the tops of the unrisen buns with the egg white glaze and sprinkle the tops with seeds or the reconstituted dried onions. Cover the buns with a clean kitchen towel and let them rise for 20 minutes. If the remaining melted butter has congealed, rewarm it just until liquefied.
Uncover the risen buns and brush the tops with melted butter (brush gently over the topping). Regardless of their shape, bake the buns in the preheated 375°F. oven until they are golden and feel light, 25 to 30 minutes. If baking in the upper and lower shelves of the oven, switch the position of the baking sheets after half the baking time. Also, check the burger buns after half the baking and, if becoming too pointy in the center, use a flat metal turning spatula to gently push this raised section down a bit, taking care not to split the sides of the bun. When done, remove the buns from the oven and brush them once more with melted butter. Transfer the buns to wire racks so they can cool until just warm. Just before serving, separate the hot dog buns by pulling them off of each other. Serve the hot dogs and burgers with your favorite mustard and relish and a bowl of spicy ketchup, all passed at the table.