Mix together the mayonnaise, honey and sriracha in a small bowl. Spread on the inside of both halves of all the buns. Divide the lettuce, tomato and onion evenly among the bottom buns.
Divide the sausage into 4 even pieces and pat and squeeze each piece into a ball. Put each ball on its own piece of plastic wrap and flatten into a thin patty 4 to 5 inches wide.
Heat the oil in a large nonstick skillet over medium-high heat. Remove the patties from the plastic wrap, add them to the skillet and cook until browned on both sides and cooked through, about 3 minutes per side. Lay a slice of Cheddar on each patty, cover the skillet and cook until the Cheddar melts, about 30 seconds. Transfer the patties to the bottom buns.
Crack the eggs into the skillet, sprinkle with some salt and pepper and cook until the egg whites are set and the bottoms and edges are crispy and the yolks are still runny, 2 to 3 minutes. Set an egg on top of each sausage patty and top with the top buns. Eat right away.
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