Combine butter and hot pepper sauce in bowl until well mixed. Set aside.
Heat gas grill on medium-high or charcoal grill until coals are ash white.
Shape pork sausage into 6 (4-inch) patties. Place onto one side of grill. Cook, turning once, 12-15 minutes or until internal temperature reaches at least 160°F. Top each sausage patty with 2 slices cheese during last minute of cooking.
Place 10x8-inch disposable aluminum or grill-safe pan onto opposite side of grill, while sausage patties are cooking. Heat pan 30 seconds. Lift pan with oven mitt; spray inside of pan with no-stick cooking spray, spraying away from open grill. Return pan to grill.
Crack each egg into pan, making sure eggs are evenly spaced apart and not breaking yolks. Cook eggs, turning once, 12-15 minutes or until egg yolks are cooked through.
Spread about 3/4 teaspoon butter over each toasted muffin half. Top bottom half of each muffin with cheese-topped sausage patty and cooked egg. Evenly divide avocado slices between sandwiches. Top with top half of English muffins.
If eggs are cooking too quickly, reduce heat or move pan away from heat on grill.