Preheat oven to 350°F. Whisk together eggs, milk, salt, pepper, and Gruyère in a medium bowl. Gently fold in green beans, Brussels sprouts, asparagus, squash, dressing, parsley, and chives.
Heat oil in a 10-inch nonstick oven-safe skillet over medium-high. Pour egg mixture into skillet. Stir once, and cook until edges start to pull away from sides of skillet, 5 to 7 minutes.
Transfer to preheated oven, and bake until set, 15 to 20 minutes. Slice into wedges; serve immediately.
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