Step 1: Preheat oven to 425 degrees. Place 2 TBSP butter in a small bowl; set aside to soften. Wash and dry all produce. Trim, peel, and cut carrots into long, thin sticks (like fries). (Ours were 4 inches long and ⅓ inch wide.) Place bacon on a baking sheet in a single layer.
Step 2: Toss carrots on a second baking sheet with a drizzle of oil, salt, and pepper. Roast carrots on top rack and bacon on middle rack until carrots are browned and softened and bacon is crispy, 18-22 minutes for carrots and 15-20 minutes for bacon. (Check often for any burning!)
Step 3: Meanwhile, halve, peel, and thinly slice onion. Heat a drizzle of oil in a large pan over medium-high heat. Add onion and season with salt and pepper. Cook, stirring, until softened and lightly browned, 5-6 minutes. Add 2 tsp sugar and ⅓ cup water (4 tsp sugar and ½ cup water for 4 servings); cook, stirring, until onion is caramelized and jammy, 3-4 minutes. Taste and season with salt. Turn off heat; transfer to a second small bowl. Wash out pan.
Step 4: While onion cooks, spread one side of each slice of bread with softened butter (if butter is not yet softened, microwave for 5-10 seconds). Place buttered sides down on your cutting board; evenly top half the bread slices with mozzarella and Pepper Jack.
Step 5: Roughly chop bacon. Sprinkle evenly on top of cheese, then top with caramelized onion. Top with remaining bread, buttered sides up, to form two sandwiches (four sandwiches for 4 servings).
Step 6: Heat pan used for onion over medium heat. Once hot, add sandwiches and cook until bread is golden brown and cheese is melted, about 5 minutes per side. (For 4 servings, you may need to work in batches.) Cut sandwiches in half on a diagonal and divide between plates. Serve with carrot fries on the side.