Prep: Preheat oven to 475°F (250°C) and set the empty pizza pan in the oven to get hot.
White Sauce: Heat 1 Tbsp oil in a large saute pan over medium heat. Add garlic and cook until soft and fragrant. Mix garlic into Greek yogurt and season with salt and pepper. Set aside.
Mushrooms: Add another tablespoon of oil to the pan, then add mushrooms in single layer. Sprinkle with salt and cook for 5 minutes. Add the sliced red onions and continue cooking until both mushrooms and onion are semi-soft, about 3 more minutes.
Assemble: Roll pizza dough out to about 12 inches wide (I like to do this directly on parchment paper for easy transfer to the hot pan later). Brush 1 Tbsp of oil onto dough, then spread on the white sauce. Top with mozzarella cheese, sauteed veggies, halved cherry tomatoes, then goat cheese. Transfer to your hot pizza pan.
Bake: Bake for 10 to 12 minutes, then move to a higher rack and broil for 2 minutes until cheese is golden brown. Serve warm, optionally topped with fresh arugula.