Sauté mushrooms in 1 Tbls. oil in a non-stick skillet over medium-high heat until browned, about 3-5 minutes.
Add garlic, cook 1 minute, then deglaze with wine; reduce until evaporated. Season with pepper and set aside
Whisk vinegar, 1 Tbl oil, Dijon, salt, and pepper together in a bowl. Add arugula and toss to coat. Set aside.
Assemble sandwiches by sprinkling ¼ cup Gouda on 2 slices of bread; top each with ½ the mushroom mixture, ½ the arugula, and another ¼ cup Gouda. Top with remaining bread.
Coat the skillet the mushrooms were sautéed in with nonstick spray. Heat the pan and 1 tsp. olive oil over medium 2 minutes.
Add the sandwiches and toast until bread is golden and cheese has begun to melt, 2-3 minutes. Turn sandwiches over and toast until bread is golden, 2-3 minutes more
Let rest 3 minutes before slicing.