Preheat oven to 425F. Lightly coat the squash with EVOO; season with nutmeg, salt and pepper. Roast until tender and browned at the edges, 18-22 minutes.
Melt the butter in a small pot over medium heat until it foams. Increase the heat a touch and add the sage leaves. Cook until crisp, then transfer sage to paper towels with a slotted spoon to drain. Chop sage leaves. Reserve brown butter.
Combine the ground veal and beef with the chopped sage and garlic. Season with salt and pepper. Mix in the Parmigiano-Reggiano and form patties.
Preheat a griddle or cast-iron skillet over medium-high heat. Add the burgers and cook 3-4 minutes on the first side, basting them with the reserved brown butter. Flip and top the burgers with cheese. Tent with foil to melt the cheese and cook for 3-4 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.