1 stick unsalted butter, cut into pieces
4 flat anchovy fillets, finely chopped
2 tablespoons drained capers, coarsely chopped
2 teaspoons fresh lemon juice
2 tablespoons finely chopped flat-leaf parsley
1 (1-pound) Italian loaf such as pane di casa (5 inches wide)
1 (1-pound) ball salted fresh mozzarella, cut into 6 (1/4-inch-thick) slices
1 cup plus 3 tablespoon whole milk, divided
1 cup fine dry bread crumbs
4 large eggs
About 3 cups olive or vegetable oil