Heat oven to 350°F. Cook mushrooms in oil in a wide skillet over medium-high heat 10 to 12 minutes until browned and liquid evaporates.
Stir in garlic 30 seconds. Add spinach by handfuls with salt and pepper; cook 3 to 4 minutes until wilted and skillet is dry.
Split connected rolls horizontally; put bottom slab in a parchment-lined 9-by-13-inch pan. Spread with Dijon; layer half the Havarti, vegetables, and remaining cheese.
Replace tops; brush with butter. Cover with foil; bake 15 minutes. Uncover; bake 8 to 10 minutes until center reaches 165°F.
Rest 5 minutes; cut into 12 sliders.