Preheat a grill to medium-high heat.
In a large mixing bowl, combine ground turkey, removed-casing turkey sausage, panko, basil, oregano, fennel seeds, garlic powder, kosher salt, black pepper, and crushed red pepper flakes.
Gently mix the ingredients with your hands until evenly combined; avoid compacting the meat.
Form the mixture into six even patties.
Grill the patties over medium-high heat until the bottoms are browned, about 5 minutes.
Flip the patties and continue grilling until cooked through, about 5 more minutes; when nearly done, top each patty with slices or shreds of mozzarella and Asiago to melt.
While the burgers cook, place the sliced onion and bell peppers in a medium bowl, drizzle with olive oil, season with salt and pepper, and toss to coat.
Put the vegetables in a grill basket and grill for about 8 minutes, turning often, until softened and lightly charred.
Toast the buns on the grill if desired.
Assemble each burger on a bun, top with the grilled vegetables and the melted cheeses, and serve immediately.