Pre-heat the oven to 425°F.
Lightly coat the squash slices with EVOO. Season with the nutmeg, salt, and peppper and spread the rings out on a baking sheet. Roast until tender and browned at the edges, 18-22 minutes.
Meanwhile, heat the butter in a skillet over medium heat until it foams. Increase the heat a touch and add the sage leaves. Cook until crisp. Transfer to paper towels to drain. Chop the sage leaves and reserve the browned butter separately.
In a large bowl, combine the veal and beef with the chopped crispy sage, the garlic, and the Parmigiano Reggiano; season with salt and pepper and mix thoroughly. Score the mixture into four equal pieces and form them into patties slightly thinner at the center than at the edges for even cooking and to ensure a flat surface (burgers plump as they cook). Drizzle the patties with EVOO.
Heat a griddle or large cast iron skillet over medium-high heat. Cook the burgers, flipping once and basting with the reserved sage butter, 10 minutes for medium (adjust the cooking time for rarer or more well done burgers). Top the burgers with Fontina during the last minute or two of cooking, tenting the pan with foil, if you like, to help melt the cheese.
Drizzle the cooked squash with honey. Place the burgers on the roll bottoms and top with the squash. Set bun tops in place.