Combine ground chicken, bread crumbs, egg, Worcestershire sauce, parsley, garlic powder, and kosher salt in a bowl using your hands; shape into 5 patties about 3 inches across and 1 inch thick.
Heat oil in a skillet over medium heat. Cook patties in the hot oil until browned, about 5 minutes. Flip patties and add onions to the skillet. Cook until patties are browned on the second side and onions are tender, about 5 minutes. An instant-read thermometer inserted into the patties should read at least 165 degrees F (74 degrees C).
Preheat oven to 425 degrees F (220 degrees C).
Mix flour, baking soda, white sugar, and salt together in a bowl; fold in chives. Cut shortening into flour mixture until crumbly. Gradually pour milk into mixture while stirring with a wooden spoon until dough starts to pull away from the sides of the bowl.
Turn dough onto a floured surface and knead for 2 minutes; flatten into 1 inch thickness. Use a round cookie cutter or juice glass to cut into individual biscuits, pushing straight down and pulling straight up to maintain fluffiness while baking. Arrange biscuits on a baking sheet.
Bake in the preheated oven until lightly browned, about 13 minutes.
Slice biscuits open and place a burger on the bottom half of each biscuit; top with onion and mozzarella cheese. Place top half of the biscuit on top of the mozzarella cheese to create a sandwich.