Cook the bacon on a griddle (or in a large skillet) over medium heat, turning occasionally, until crisp, 8 to 10 minutes.
Meanwhile, whisk the eggs and sour cream in a medium bowl. Remove the bacon from the griddle and drain on paper towels; pour the fat into a small bowl and set aside.
Add the egg mixture to the griddle and cook, stirring with a rubber spatula, until just set. Layer 1 slice of cheddar, some scrambled eggs, a piece of bacon and another slice of cheddar on each of 4 slices of bread. Top with the remaining bread.
Return the bacon fat to the griddle; add the butter and let melt over medium heat. Place the sandwiches on the griddle, cover and cook until the bread is golden brown and the cheese melts, about 4 minutes per side.