Toast the bread and rub each slice with the garlic. Drizzle the toast with the olive oil. Set the toast on a plate and arrange the radicchio leaves on the toast.
Put 1 tablespoon of the butter in a heavy, medium nonstick skillet and place over medium-low heat, swirling the skillet until the butter is just melted. Crack the eggs directly into the warm pan; poke the yolks, then stir in the milk. (This method eliminates an extra step-and means fewer dirty dishes-but you can also crack the eggs in a bowl and whisk them together with the milk until blended.) Stir the eggs every minute or so, with a wooded spoon or silicon spatula. When they begin to get lumpy, sprinkle with the salt and stir slowly until they begin to form a glistening, creamy mound. Stir in the remaining 1/2 tablespoon of butter and spoon the eggs onto the radicchio.
Shred a small handful of cheese over the eggs and scatter the parsley on top.