Puree pumpkin flesh in a blender or food processor with 1-2 tablespoons water or broth.
Combine all sauce ingredients with pumpkin in a medium pot and simmer over medium heat for 3-5 minutes. Stir occasionally. Once warmed through, Remove from heat and keep covered.
Preheat grill for medium heat.
Cut pizza dough in half and roll each into 8” circles. Place dough onto the grill grates and cook for about 2-3 minutes, until the dough hardens a bit and turns golden. DO NOT TURN over. Remove dough from grill and place the raw side down onto a parchment-lined baking sheet.
Slather cooked side with pumpkin sauce and return dough back onto the grill raw side down and cook for 4 minutes or until dough has cooked through. Promptly remove pizza and top with arugula, sliced tomatoes, and shaved parmesan. Slice and serve.