Heat oven to 400°F. Arrange bacon on a rack over a rimmed sheet pan and bake for 16 to 20 minutes until crisp, then transfer to a clean plate.
Mix cheddar, cream cheese, chopped pepper and smoked paprika into a thick spread. Whisk eggs with sour cream, salt and black pepper until mostly smooth.
Reduce oven to 300°F and warm the split croissants cut-side up for 5 minutes. Spread the pimento cheese over their bottom halves.
Melt butter in a large nonstick skillet over medium-low heat. Add eggs and draw a spatula slowly across the pan for 4 to 6 minutes, forming soft curds, until set and 160°F.
Divide scrambled eggs over the croissant bottoms. Break bacon slices only as needed to fit and add two slices to each sandwich.
Set the croissant tops over the filling without pressing and serve while the eggs are warm and the bacon is crisp.