Position an oven rack in the center of the oven and preheat to 400 degrees F.
Brush 6 standard muffin cups with 2 tablespoons of the olive oil; the oil will pool on the bottom of the cups, but that's okay. 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 until the egg whites are set and wobble only slightly, 8 to 10 minutes (the eggs will continue to cook after they come out). Let cool 5 minutes, then use an offset spatula to loosen the prosciutto from the edges of each cup and lift onto a plate for serving.
Mix the basil, walnuts, red pepper flakes and garlic with the remaining olive oil in a small bowl to combine. Serve over the egg cups and remaining tomato wedges.
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