Preheat the oven to 375 degrees F (190 degrees C). Heat oil in a large skillet over medium-high heat. Season chicken pieces well with garlic powder, salt, pepper, chili powder, and cumin.
Add chicken to the hot skillet and cook, undisturbed, until browned and easily releases from the pan, 3 to 4 minutes.
Add onion and cook, stirring occasionally, 3 more minutes. Add 2/3 cup salsa, beans, and green chiles and bring to a simmer. Reduce heat slightly and cook, stirring occasionally, until slightly thickened and creamy, about 5 minutes.
Brush sides and bottom of an 11x7-inch baking pan or baking dish with butter. Working with 1 tortilla at a time, use remaining butter to brush 1 side of each tortilla and place in the baking dish. Place about 1 tablespoon cheese in bottom of a tortilla and top with about 1/3 cup filling and about 2 more tablespoons cheese. Fold tortilla up like a taco and place it against the edge of the pan.
Place a second buttered tortilla against the next one and top with cheese and filling. Fold in half and repeat with remaining tortillas, filling, and cheese to fill the baking dish with side-by-side tacos. Top with any remaining cheese.
Bake in the preheated oven until the tops of each tortilla are golden brown and toasted and cheese is melted and bubbly, 20 to 25 minutes.
Meanwhile, stir together remaining salsa with sour cream until well combined. Top tacos with lettuce just before serving and drizzle with sour cream sauce.
