Preheat the oven to 400°F (200°C).
In a large bowl, gently toss the cauliflower slices and the onion rounds with 1 tablespoon olive oil until evenly coated. Arrange them in a single layer on a large baking sheet and season with salt and freshly ground black pepper to taste.
Roast in the preheated oven for 20 minutes, or until the cauliflower is golden brown and tender. (Turn the pieces once about halfway through if you want more even browning.) Remove from the oven and let cool to room temperature.
Meanwhile, make the aioli: in a small bowl stir together 1/4 cup light mayonnaise, 1 small garlic clove (minced), 2 teaspoons whole grain mustard, and 2 teaspoons fresh lemon juice. Season with salt and freshly ground black pepper to taste.
Spread the 2 tablespoons softened butter on the outsides of the 4 slices of whole-grain bread. Spread the aioli on the inside surfaces of all 4 slices.
Assemble two sandwiches: on the aioli side of two bread slices, place the cheese and fillings in this order—one slice Arla Dofino® Gouda (or the equivalent portion if using 2 oz grated), then half of the roasted cauliflower and onion, then one slice Arla Dofino® Creamy Havarti (or the equivalent portion if using 2 oz grated), and then one-third cup total baby arugula divided between the two sandwiches (i.e., about 1/6 cup per sandwich). Top each with the remaining bread slice, aioli side down (buttered side out).
Heat a large skillet or griddle over medium heat until hot. Place the sandwiches in the skillet, buttered side down.
Cook the sandwiches until the bottom bread is golden brown and the cheese begins to melt, about 4–6 minutes. Carefully flip each sandwich and cook the other side until golden brown and the cheese is fully melted, about 3–4 minutes more.
Remove the sandwiches from the skillet, cut each in half, and serve immediately.