Preheat the oven to 450°F. Toss the potatoes on a baking sheet with the oil, salt, pepper, and rosemary. Roast until fork-tender, but not mushy, about 15-18 minutes.
To make the aioli, combine all the ingredients in a small bowl. Stir well with a whisk and season to taste.
For the burgers, combine the lemon, parsley, garlic, oil and salt in a blender until mostly smooth to make a gremolata.
Toss the mushrooms with the gremolata.
Heat a grill or large skillet to medium heat with a drizzle of oil.
Sear the mushrooms on both sides until browned and softened.
Brush the onions with balsamic and olive oil and grill/sear on both sides until browned and softened, 3-4 minutes per side.
Serve the grilled mushrooms on hamburger buns topped with a slice of gouda, a seared onion ring, arugula and rosemary aioli. Pair with rosemary roasted potatoes.