Cook sausage patties in a lightly oiled skillet over medium heat for 3–4 minutes per side until browned and cooked through; drain on paper towels. (If using bacon, cook until crisp.)
Lightly butter the cut sides of the English muffins. Toast them in the same skillet, cut-side down, for 2–3 minutes until golden; place a slice of cheese on each bottom half to begin melting.
Grease a round egg ring (or a clean Mason jar lid) and place it in the skillet over medium-low heat. Add a teaspoon of butter and a splash of water to the pan, crack an egg into the ring, season with salt and pepper, cover, and cook 2–3 minutes until whites are set and yolk is cooked to your liking. Repeat for remaining eggs.
Assemble: place a sausage patty (or bacon) on each cheese-topped muffin half, top with a cooked egg, and add optional hash brown patty if using. Cap with the top muffin half.
Serve immediately, or wrap tightly and freeze for up to 2 months. Reheat from frozen in the microwave 60–90 seconds or in a 350°F oven for 10–12 minutes.