Preheat a grill to medium-high. While the grill heats, make the burgers by placing the ground turkey, turkey sausage, bread crumbs, basil, oregano, fennel, garlic powder, salt, black pepper, and red pepper flakes in a large mixing bowl.
Gently mix with your fingers. Make sure the ingredients are fairly evenly incorporated, but do not compact the meat.
Shape into 6 patties, then grill on medium high until the bottoms are browned (about 5 minutes).
Flip and continue grilling until cooked through; about 5 additional minutes. When the burgers are almost ready, top with mozzarella and Asiago cheeses.
Meanwhile, place the veggie slices in a medium bowl. Drizzle with olive oil, then sprinkle with salt and pepper. Toss to coat.
Place in a grill basket and grill until softened and lightly charred, about 8 minutes, turning the veggies often so that they do not burn. Toast the buns as desired.
Serve burgers on the buns, topped with grilled veggies. Enjoy immediately.
TO STORE: Leftover cooked turkey patties may be stored in a container with a tight-fitting lid in the refrigerator for up to 3 days.
TO FREEZE: Uncooked patties may be frozen on a baking sheet until firm, then transferred to a freezer-safe container for storage for up to 3 months. To cook, defrost overnight in the refrigerator, then grill as directed. Cooked patties may be kept frozen for up to 3 months.
TO REHEAT: If frozen, defrost cooked turkey burger patties in the refrigerator overnight. Reheat in a skillet, covered, with a little extra water to help the burgers stay moist.