Get Ready: Preheat your oven to 350°F (180°C). Lightly grease four large (Texas-sized) muffin cups with melted butter.
Cook Bacon: In a non-stick pan over medium-high heat, cook the chopped bacon until it's lightly crispy. Drain any excess fat on paper towels and set aside.
Mix Dry Ingredients: In a large bowl, whisk together the flour, baking powder, and baking soda until well combined.
Mix Wet Ingredients: In a separate bowl, whisk together the milk, white vinegar, salt, sour cream (or yogurt), vegetable oil, and the single egg until smooth.
Combine Batters: Pour the wet ingredients into the dry mixture. Stir gently with a spoon or spatula, just until the flour is mostly incorporated. Don't overmix to keep your muffins tender!
Fold in Flavor: Add the cooked bacon, sliced green onions, and shredded cheese to the batter. Give it a few gentle stirs to evenly distribute these delicious additions.
Assemble Muffins: Spoon about ¼ cup of batter into each of the four prepared muffin cups. Use your spoon to create a small indent or well in the center of the batter, then carefully crack one whole egg into each well.
Cover Eggs: Divide the remaining batter among the muffin cups. The easiest way to do this is to dollop batter around the egg yolk first, then on top of the yolk, and gently spread it to completely cover the egg.
Bake: Brush the tops of the muffins with the remaining melted butter. Bake for about 25 minutes, or until the muffins are golden brown and cooked through.
Serve Warm: Remove the muffins from the oven and let them rest in the pan for about 5 minutes before carefully turning them out onto a cooling rack.