Cut onion in half, then slice each half into 6 slices. In a large skillet or on a griddle, sauté onions in oil until soft, about 5 minutes, stirring occasionally. Remove onions; keep warm.
Wipe out pan, coat with cooking spray. Add veggie patties; cook over medium heat until browned and cooked through, about 5 minutes per side. Remove patties from pan/griddle; keep warm.
Fry eggs in small batches in spray-coated pan on first side; turn over and continue cooking for about 2 to 3 minutes over medium heat until whites are set (completely coagulated and firm) and yolks begin to thicken (no longer runny, but not hard).
Blend mayonnaise and hot pepper sauce.
For each sandwich, spread 1 tablespoon mayonnaise on English muffin bottom. Top with a vegetable patty, layer with egg, a slice of cheese, and 1 tablespoon onions. Add muffin top. Serve immediately.