Whisk together sour cream and flour in a small bowl.
Cook scallions, butter, and ¼ tsp each salt and pepper in a skillet until scallions are tender. Add cream and simmer until just thickened, about 2 minutes. Stir in sour cream mixture and cook 1 minute. Remove from heat and stir in ⅓ cup cheese. (This can be made 3 days ahead and kept chilled until you’re ready to serve.)
Heat oil in a large skillet until hot. Add mushrooms, season with salt and pepper, and sauté until golden, about 5 minutes. Add garlic and cook 1 minute longer. Stir in parsley and remove from heat.
Toast bread. Set oven to broil. Arrange toasts on a baking sheet and generously spread with scallion-cream and sprinkle with remaining cheese. Broil until brown and bubbly. Top with mushrooms.