Position the oven racks in the upper-and lower-third shelves and preheat the oven to 375°F. Use some of the melted butter to generously grease two cushioned cookie sheets. Reserve any remaining melted butter. Pour ¾ cup granulated sugar on a doubled paper towel that sits on your counter. Use some of the reserved melted butter to generously grease the bottom of the outside of a wide drinking glass, and plant the buttered surface directly in the sugar, pressing down, and leave it in the sugar until needed. Put four wire cooling racks on your counter and place two long overlapping sheets of waxed paper under two of the racks (these will be for the tops of the cookies).
Use an electric mixer with a paddle attachment, or a handheld mixer, to cream the 1 cup softened butter with the remaining 1 cup of sugar until light and creamy. Add the peanut butter and combine well. Turn off the machine and add the flour. With the machine on the lowest setting, mix the dough until fully combined, without overworking it. Form walnut-size balls of dough and place them 1½ inches apart on the prepared cookie sheets. (You should be able to bake 9 to 12 cookies on each sheet, if using a 2½-inch cookie cutter.) Lift the buttered and sugared glass and press down on each ball of dough, flattening it to ¼ inch thick. After flattening each portion of dough, rub the greased glass in the sugar before flattening the next. (If the sugared surface gets clumpy, wipe it off and reapply some melted butter and sugar.) Use a clean pastry brush to gently remove any excess sugar from the tops of the flattened cookie dough.
Use a 2½-inch heart-shaped cookie cutter to cut out hearts from the flattened dough. Before removing the cutter, use your finger to remove the excess cookie dough, and place these scraps back in the original bowl. Insert a small cutter or a thimble into half (or a few more) of the shaped cookies and rotate the cutter in a circular motion to widen the hole slightly. Lift out the cutter and, using the pointed tip of a short paring knife, pick the dough out of the hole. Put these pieces of dough back into the bowl with the other scraps.
When the cookie sheets are full, transfer them to both levels of the oven and bake for 6 to 9 minutes, switching their positions after half the baking time. Check after 6 minutes. When done, the cookies will be slightly more golden around the edges but the tops should be only a light golden color. Remove the sheets from the oven and place the sheets with cookies with holes on the wire racks set over waxed paper. Place the sheets with uncut cookies on the two other racks and let them all cool for 10 minutes. Using a very thin metal turning spatula, carefully transfer the cookies to their wire racks to cool further. (Wipe off the spatula after removing each cookie from the sheet.) Cool, clean, and grease the cookie sheets and repeat the above procedure until all of the cookie dough has been used.
Once completely cool, assemble the sandwiches. First, generously sift the confectioners’ sugar over the cookies with holes. Lift and turn a hole-free cookie flat side up, and spoon a generous teaspoon of jam in the center. Carefully lift a top “sugared” cookie and (gently now) lay it, sugared side up, on top of the jam, applying only the slightest bit of wiggling pressure so the jam plumps up through the hole. Place the assembled sandwich cookie back on its rack while you continue to assemble the rest.
If serving soon, use a thin metal spatula to transfer the cookies to a decorative serving platter. If serving later in the day of baking, or the next day, carefully transfer the cookies to a deep rectangular container (like a large roasting pan) to protect their appearance. Cover the pan securely with aluminum foil and leave it at room temperature.
Melt 12 ounces of semisweet or milk chocolate in either the top of a double boiler over barely simmering water, stirring constantly, or in a microwave (uncovered) for 1 minute on high power, stirring after heating until smooth. Apply the chocolate as you would jam to the flat side of a cookie and sandwich them as directed.