Chop sage and garlic (or grate garlic on microplane) then in a medium bowl combine the maple syrup, sage, garlic, 1 teaspoon salt, paprika, onion powder, cayenne pepper, and black pepper. Add the pork and mix until thoroughly mix until fully combined, do not overmix. Transfer to a zip-top bag, or non-reactive container, with a cover; refrigerate 2 hours and up to 2 days.
Form Sausage Patties:
Using your hands, form the sausage mixture into 5-inch patties, about 1/3- inch thick.
Heat a large cast-iron pan or griddle over medium-high until very hot, 6 to 7 minutes. Grease skillet with 1 tablespoon of vegetable oil.
Add the the sausage patties, gently pressing down with a spatula, in order to adhere to the pan. Cook, undisturbed, until bottom of sausage is dark brown and crisp, 2 to 3 minutes. Flip patties, using spatula to “smash” lightly again; and sausage is dark golden brown on other side,and fully cooked through, 2 to 3 minutes more. Transfer sausage to a plate and cover to keep warm.
Remove skillet from heat. Using paper towels, wipe any excess oil, or sausage bits from the skillet. If skillet is smoking hot allow to cool slightly.
Replace skillet on heat, over medium; add remaining 2 tablespoons of vegetable oil and sprinkle with remaining teaspoon salt.
Gently crack eggs into a bowl, being careful not to break yolks. Lower bowl with eggs close to skillet. Carefully pour eggs, one at a time skillet, trying to spacing out evenly. Cook eggs until just set on the bottom 1 to 2 minutes. Using a thin spatula (you may have to use the spatula to slice the egg whites between the eggs to separate them). Carefully flip each egg and top each with a slice of cheese. Cover skillet with a baking sheet or pot lid large enough to cover eggs. Continue to cook until cheese has melted and yolks are still runny, or cooked to your preference 1 to 2 minutes more.
Split and toast sandwich buns. Dividing evenly, layer buns with eggs, and top each with sausage patties. Top each with bun tops and serve hot.