In a large nonstick skillet over medium-high heat, cook bacon until browned, about 4 minutes per side. Drain slices on paper towels, then crumble. Reserve rendered fat in skillet for hash browns, if desired.
Heat broiler. In a medium bowl, whisk together eggs and milk, then whisk in the remaining ingredients (except muffins and 4 tablespoons cheese).
Heat an ovenproof 10-inch nonstick skillet over medium-low heat. Pour in the egg mixture, and cook until bottom is set, about 7 minutes. Transfer to broiler, and cook until top is set and puffy, about 1 minute. Remove from oven, and immediately slide onto a cutting board; let rest 1 minute.
Using a biscuit cutter or inverted glass, cut out four rounds, or cut into four wedges with a knife. Place one portion on the bottom half of each of the English muffins. Transfer to a baking sheet.
Sprinkle remaining 4 tablespoons cheese over the sandwiches; broil until cheese is melted and bubbling, about 30 seconds. Place top half of muffins on each sandwich, and serve immediately.