Preheat oven to 400°F. Toss the vegetables with olive oil and dill and season with salt and pepper. Spread out on a baking sheet and roast for 10 minutes, until vegetables are slightly tender.
Brush the melted butter on one side of the bread slices.
In a large cast iron skillet or electric griddle over medium-low heat, place one slice of bread on the skillet, buttered side down. Add half the Dill Havarti cheese on top of the bread, then layer with half the roasted vegetables, followed by half the Gouda cheese. Top with a bread slice, buttered side up.
Cook the sandwich until golden brown and cheese is melted, about 4-6 minutes on each side. Repeat for the second sandwich.