1. Trim and discard root ends and tough tops from leeks; peel 1 layer off each leek. Cut leeks in half lengthwise and hold each half under cool running water, flipping to remove grit between layers. Thinly slice crosswise (you should have about 3 cups). Scrub potatoes and cut into 1/3-inch chunks.
2. In a 10- to 12-inch nonstick frying pan (with ovenproof handle), combine leeks, potatoes, olive oil, thyme, and 1 cup water. Bring to a boil over high heat; cover pan, reduce heat to medium, and simmer, stirring occasionally, until potatoes are tender when pierced, 6 to 8 minutes. If any liquid remains in pan, uncover and simmer, stirring often, until evaporated. Spread vegetable mixture level in pan.
3. Meanwhile, in a bowl, beat eggs, egg whites, 1/2 teaspoon salt, 1/4 teaspoon pepper, and 1 tablespoon water to blend. Pour egg mixture over vegetables in pan. As edges begin to set, lift with a spatula and tilt pan to let uncooked egg mixture flow underneath. Cook until eggs are firm but still moist on top, 2 to 4 minutes.
4. Remove from heat and sprinkle with cheese. Broil about 6 inches from heat until cheese is melted, 1 to 2 minutes. Sprinkle with parsley. Slide frittata onto a plate; cut into wedges. Add salt and pepper to taste. Serve with toast.