Preheat the broiler and place the top rack 4 to 5 inches from the broiler element. In a large bowl, whisk together the eggs, cream, 1/4 teaspoon of the salt, and 1/8 teaspoon pepper until the eggs are frothy. Add 1/2 cup of the cheese and whisk. Set aside.

In a 10-inch nonstick ovenproof skillet, melt the butter over medium-high heat. Add the zucchini, and mushrooms, and and cook, stirring, until the vegetables are soft and lightly browned, 2 to 3 minutes. Add the peppers and green onions, and cook, stirring occasionally, for 2 minutes. Add the garlic, thyme, 1/4 teaspoon of the salt, and pinch of the pepper and cook, stirring, until the garlic is fragrant, 30 seconds to 1 minute. Stir in 1 tablespoon of parsley and with the pan over medium heat, pour the egg mixture over the vegetables. Cook, lifting with a rubber spatula to let the eggs flow underneath, until the edges are set but the middle still is loose, 3 to 4 minutes.

Remove from the heat and sprinkle the remaining 1/4 cup cheese over the top. Place under the broiler until the eggs are slightly puffed and the cheese is bubbly and golden brown, 2 to 3 minutes, watching carefully to prevent from overcooking.

Remove from the oven and carefully slide the frittata out onto a large serving platter. Garnish with the remaining 1 tablespoon parsley and the chopped green onions. Slice, and serve immediately.