Over a medium bowl, grate 2 cloves garlic. Stir in the mayonnaise, one-third of the anchovies, 1 tablespoon lemon juice and the capers. Mix in the tuna, flaking the fish with a fork.
Using a blender, puree the remaining anchovies, lemon juice and garlic clove until smooth. With the machine on, slowly pour in 1/4 cup olive oil. Transfer to a large bowl and toss with the frisee and parmigiano-reggiano; season with salt and pepper.
Split the rolls. Using a 2 1/2-inch round cookie cutter, cut a hole from the middle of each roll top. In a medium nonstick skillet, heat 2 tablespoons olive oil over medium heat; add the roll bottoms cut side down and cook until toasted. Transfer cut side up to 4 plates. Add the remaining 2 tablespoons olive oil to the pan; add the roll tops cut side down, then crack an egg into each hole. Lower the heat to low, season with salt and pepper and tent the skillet with foil. Cook until the eggs are set, about 5 minutes.
Divide the frisee salad among the roll bottoms, top with the tuna salad, then set the roll tops in place.