Preheat oven to 325°F.
In a shallow bowl, combine plain breadcrumbs, panko, garlic powder, dried parsley, salt, and pepper; set aside. Place flour on a plate. In a separate bowl, beat the egg with 1 tablespoon of water.
Using a meat mallet, gently pound each chukar breast to tenderize.
Heat olive oil in a large skillet over medium-low heat. Dredge each breast in flour, dip into the egg wash, then press firmly into the breadcrumb mixture.
Fry chukar breasts until golden brown, about 1 minute per side. Transfer to a baking sheet, top each breast with your choice of cheese, and bake for 15-20 minutes, until the cheese is melted.
Meanwhile, butter the brioche buns and toast them in the oven for 5 minutes. Spread optional mayonnaise on the buns if desired.
Assemble the sandwiches by layering a slice of tomato on the bun, then topping with the chukar, jalapeño slices, and red onion. Serve immediately while hot.