Preheat the oven to 425 degrees . Lightly coat the squash slices with EVOO; season with nutmeg, salt and pepper. Roast until tender and browned at the edges, 18 to 22 minutes.
Melt the butter over medium heat until it foams. Increase the heat a touch and add the sage leaves. Cook until crisp, then transfer them with a slotted spoon to paper towels to drain, reserving the brown butter. Chop the sage leaves.
Combine the ground veal and ground beef with the sage and garlic; season with salt and pepper. Mix in the parmigiano-reggiano and form patties.
Heat a griddle or cast-iron skillet over medium-high heat. Add the burgers and cook, basting with the reserved brown butter, for 3 to 4 minutes. Flip, top the burgers with the fontina (or taleggio) to melt and cook for another 3 to 4 minutes.
Drizzle the cooked squash with honey. Lightly toast the buns in a warm oven. Place the cheeseburgers on the roll bottoms, top with the squash rings and set the roll tops in place.