Gently toss the cauliflower and onion rounds with olive oil. Place on a large baking sheet and season with salt and black pepper, to taste. Roast for 20 minutes or until cauliflower is golden brown and tender. Remove from oven and set aside to cool to room temperature.
To make the aioli, add the mayonnaise, garlic, mustard, lemon juice to a small bowl. Stir until combined. Season with salt and black pepper, to taste.
Butter the outsides of the bread with butter. Spread the insides of the bread slices with aioli. Layer shredded Gouda cheese, roasted cauliflower, onion, sliced Havarti cheese, and arugula. Top with the other slice of bread.
Heat a large skillet or griddle pan over medium heat and place the sandwiches butter side down. Cook for 4-6 minutes, or until the cheese starts to melt and the bread is golden brown. Carefully flip the sandwich over and cook for an additional about 3-4 minutes. Cut sandwiches in half and serve immediately.