Mix chili powder, 1 1/2 teaspoon salt, cumin, minced garlic, and black pepper together in a small bowl.
Sprinkle 3/4 of the chili powder mixture over ground chicken in a large bowl. Add egg; mix well and shape into 4 patties.
Preheat grill for medium heat and lightly oil the grate. Grill patties until browned and juices run clear, about 5 minutes per side. An instant-read thermometer inserted into the center should read at least 165 degrees F (74 degrees C).
Mash avocado in a small bowl; season with garlic powder and 1/4 teaspoon salt.
Mix remaining chili powder mixture, sour cream, and shredded cabbage together in a bowl to make coleslaw.
Toast Kaiser rolls lightly on the grill. Place chicken patties on the buns. Top each patty with mashed avocado, a large scoop of coleslaw, and sliced tomato.