Heat oven. Heat oven to 350°F and grease a 9 x 13-inch pan.
Make syrup. Combine water, piloncillo (or brown sugar), canela, raisins, and cloves in a large saucepan. Bring to a boil then reduce heat to medium-low. Whisk occasionally until piloncillo is dissolved. Simmer, uncovered, until raisins are plump and syrup has thickend slightly, about 20 minutes.
Toast bread. Slice bolillos into 1-inch thick slices. Lay in a single layer on a baking sheet and brush both sides with the melted butter (use all the butter). Bake until crisp, about 20 minutes, flipping the bread slices halfway through.
Layer. Discard canela and cloves in the syrup. Lay half of the bread slices in a single layer on the bottom of the baking dish. Top with half the cheese. Ladle half the syrup over the cheese and bread. Repeat with the other half of the bread, cheese, and syrup.
Bake. Grease a large piece of aluminum foil then cover the Capirotada (greased side down) and bake for 30 minutes. Remove the foil and bake another 10 minutes or until browned on top.
Serve. Sprinkle with powdered sugar if desired and serve warm.