Preheat oven to 275° F.
Combine all ingredients for the rub. Rub pork butt generously with Ancho Chile Rub.
Quickly sear on all sides in a nonstick oil-coated sauté pan over medium heat until dark brown.
Place in roasting pan, fat side up. Cover with foil and seal.
Roast for 4 ½ -5 hours at 275° F until tender, and internal temperature reaches 180° F. Remove from pan and rest on a cutting board.
Cool slightly, then shred with a fork. Keep warm.
Whisk together flour, baking powder, baking soda, sugar and salt.
Mix in frozen grated butter and grated cheese.
Stir in buttermilk and form a dough, bringing it together gently and kneading only once or twice.
Roll out until dough is 1-inch thick and cut into twelve 3-inch rounds.
Bake at 375° F for about 18 minutes, or until golden brown on the bottom. Keep warm.
For each egg patty, whisk one large egg (or about ¼ cup liquid whole eggs) with 1 teaspoon minced chives.
Pour into egg ring on preheated griddle or sauté pan that has been seasoned with nonstick spray.
Cook until edges of the egg are set, then remove ring and carefully flip over to cook the other side until egg is completely firm.
Split a warm biscuit and cover the bottom with one egg patty, 2 oz. pulled pork, 1 slice cheese, 2 tablespoons caramelized onions and 2 tablespoons avocado cream.
Cover with top of biscuit and serve immediately.
*If using frozen or liquid whole egg product.
NOTE: Do not allow raw or cooked eggs to remain at room temperature for longer than one hour (including preparation and service time).