How to Make It
Preheat the broiler. Brush both sides of the brioche slices with softened butter and arrange in a single layer on a large rimmed baking sheet. Broil the brioche 8 inches from the heat, turning once, until lightly toasted, about 2 minutes total. Leave the broiler on.
In a medium bowl, toss the frisée with the parsley, scallions, lemon juice and olive oil. Season the salad with salt and pepper.
Melt 1 tablespoon of butter in each of 2 large nonstick skillets. Crack 3 eggs into each skillet and cook sunny side up over moderate heat, until the whites are firm and the yolks runny, about 5 minutes. Transfer to a plate and season with salt and pepper.
Top the brioche with the Gruyère, covering as much of the toasts as possible with the cheese. Broil 8 inches from the heat for about 3 minutes, until the cheese is melted. Transfer the cheesy brioche toasts to plates. Top with the frisée salad, prosciutto and fried eggs and serve at once.