Preheat the oven to 350 degrees F (175 degrees C).
Combine water, piloncillo, cinnamon sticks, sugar, vanilla extract, and cloves in a pot. Bring to a boil. Reduce heat and cook until slightly thickened and reduced to a syrup, about 10 minutes. Remove cinnamon sticks.
Lightly butter each slice of bread. Place slices in a large, flat baking pan.
Bake in the preheated oven until toasted, 10 to 15 minutes. Let butter dry.
Grease a 9x13-inch cake pan and add 1/2 of the toasted bread. Sprinkle 1/2 of the Monterey Jack cheese, walnuts, raisins, and apricots over the bread. Drizzle 1/2 of the syrup on top to fully coat the bread. Top with the remaining bread, cheese, walnuts, raisins, apricots, and syrup. Cover with aluminum foil.
Bake in the preheated oven for 20 minutes. Remove foil and continue baking until lightly browned and heated through, about 15 minutes more. Serve warm.
