Preheat the oven to 400 degrees F with an oven rack in the middle.
Brush 2 tablespoons of olive oil into 6 regular muffin cups; the oil will pool at the bottom of the cups, which is fine. Line each cup with a slice of prosciutto, folding and overlapping so that the entire surface of the cup is covered and no metal is peeking through. Crack 1 egg into each prosciutto cup and top each egg with a tomato wedge. Reserve the remaining tomato wedges for serving.
Bake for 8 to 10 minutes, or until the egg whites are set and barely slightly jiggle (the eggs will continue to cook after they come out). Allow 5 minutes to cool before using an offset spatula to release the prosciutto from the cup's borders and lift onto a plate to serve.
In a small bowl, combine the basil, walnuts, red pepper flakes, and garlic with the remaining olive oil. Serve with the leftover tomato wedges and egg cups.