1. Preheat the oven to 350 degrees. If you have a convection oven, set it on convect.
2. Cut open the pumpkin from top to bottom. Scrape out all the seeds and fibers. Cut the pumpkin into eighths. Carve the meat out in pieces as large as you can, leaving ½-inch-thick shells. Cut the meat into 1/8-inch-thick slices.
3. Crush the 3 whole garlic cloves and use them to wipe the inside of a 12 x 8-inch glass baking dish. Discard what’s left of the crushed cloves. Shingle half of the pumpkin and half of the carrots along the bottom of the dish. Sprinkle on half of the chopped garlic, one-third of each of the cheeses, and white pepper to taste. Repeat the process with the remaining ingredients, finishing with a generous layer of the cheeses.
4. Beat the half-and-half, egg yolks, and nutmeg together, then pour over the casserole. Wrap a relatively tight seal of aluminum foil over the top of the dish. Bake for 1 hour and 10 minutes.
5. Raise the oven temperature to 400 degrees. Remove the foil, sprinkle bread crumbs in a thin layer over the top, and return to the oven. Continue baking, uncovered, until the crust browns.
6. Remove from the oven and allow to rest and cool for at least 15 minutes before serving.