In skillet, cook onion, cremini, and garlic in 2 tbsp. oil over medium-high, stirring often, until browned, about 15 minutes; season. Brush portobellos with remaining 2 tbsp. oil. In grill pan, cook portobellos, stemside down, over medium-high for 5 minutes. Turn; top with cremini mixture, then cheese. Cook until tender, about 5 minutes. Spread buns with Dijonnaise. Add arugula and portobellos.