To make a bed for these hors d'oeuvres, grate 5 to 7 ounces Parmesan in long, thin strips. Guests can also use this cheese to sprinkle over the cups, if they like.
Heat oven to 375 degrees with rack in upper third. Using a fork, combine butter, cayenne, garlic paste, salt, and pepper in a bowl. Reserve at room temperature.
Cut crusts off bread. On dry surface, flatten one slice of bread with rolling pin to 1/16 inch. Cut circle with a 2 1/2-inch-round cookie cutter. Brush butter mixture onto one side of bread. Press bread round, buttered-side up, into hole of 24 minimuffin tin (each cavity measuring 2 inches in diameter and 3/4 inches deep), letting bread scallop inside edges of each tin. Repeat, with slice of bread at a time, until all are used.
Bake until bread is toasty, 6 to 8 minutes. Transfer to baking rack to cool and set. May be made a day ahead and kept in an airtight container at room temperature.
Just before serving, stack lettuce, and cut across into 1/8-inch-thick strips. Transfer to a medium bowl. Toss with Caesar dressing just before serving so lettuce does not wilt. Fill each bread cup with lettuce; top with frico topping and minicroutons.