Preheat the oven to 400°F (200°C).
Toss the sliced cauliflower and onion rounds with the olive oil in a bowl until evenly coated; season with salt and pepper.
Arrange the cauliflower and onion in a single layer on a large baking sheet and roast for about 20 minutes, until the cauliflower is golden and tender; remove and let cool to room temperature.
Make the aioli by combining the mayonnaise, minced garlic, whole grain mustard, and lemon juice in a small bowl; stir to combine and season with salt and pepper to taste.
Butter the outsides of each bread slice with the softened butter. Spread the aioli on the inside of each slice.
Assemble each sandwich: layer shredded Gouda, roasted cauliflower and onion, sliced or shredded Havarti, and baby arugula on one bread slice, then top with the other slice, aioli side down.
Heat a large skillet or griddle over medium heat and place the sandwiches butter-side down. Cook until the bottom is golden and the cheese begins to melt, about 4–6 minutes.
Carefully flip each sandwich and cook an additional 3–4 minutes until the second side is golden and the cheese is fully melted. Remove, cut in half, and serve immediately.