For the Chicken
3 slices white sandwich bread, torn into pieces
2 tablespoons unsalted butter, cut into small pieces and chilled
1 tablespoon finely grated lemon zest
1 small garlic clove, minced
6 tablespoons nonpareil capers, drained
Salt and pepper
1/3 cup all-purpose flour
2 large eggs, at room temperature
4 thin-cut chicken cutlets
3/4 cup olive oil

For the Mayonnaise
1 large egg yolk, at room temperature
2 teaspoons Dijon mustard
2 teaspoons lemon juice
Salt and pepper
2/3 cup vegetable oil

For the Assembly
4 sturdy rolls, split in half
2 tablespoons unsalted butter, softened