In a large bowl, thoroughly mix the minced chicken, minced beef, galangal powder, minced lemongrass, sliced kaffir lime leaves, minced garlic, fish sauce, yellow curry paste, and black pepper. Form into 4 equal patties.
In a small saucepan over medium-low heat, combine the peanut butter, coconut milk, soy sauce, and 1 teaspoon of palm sugar. Whisk continuously as it simmers until the sauce renders smooth and visibly thickens. Remove from the heat and set aside.
In a mortar and pestle, crush the garlic clove and pepperoncino. Add the roasted peanuts, 2 teaspoons of palm sugar, and 1 tablespoon of fish sauce, pounding into a rough dressing. Toss with the julienned mango and cucumber.
Toast the brioche buns on both sides until golden brown.
Heat 1 teaspoon of neutral cooking oil in a skillet over high heat. Lay the patties down and cook on the first side until deeply seared. Flip them over, immediately top each individual patty with a slice of chili cheddar cheese, and let the remaining side finish cooking so the cheese melts completely.
Assemble by spreading garlic sauce or mayonnaise on the bottom bun (if using). Layer the cheesy patty, a generous handful of the mango cucumber salad (leaving the excess liquid behind in the bowl), sliced green onions, a heavy drizzle of the warm peanut sauce, and the top bun.