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Preheat oven to 350 F. Melt the butter in a large nonstick skillet over medium. Add the bell peppers and onions, sauté until tender, 4 to 5 minutes. Stir together eggs, salt, pepper and water in a medium bowl. Add the egg mixture to the pepper mixture, and cook, without stirring, until the eggs begin to set on the bottom, about 1 to 2 minutes. Draw a spatula across the pan to form large curds. Cook, stirring occasionally, until the eggs are thickened about 6 to 7 minutes. (Do not overstir.)
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Spread 2 tbsp of the salsa verde in the center of each tortilla. Spoon about ¼ cup of the egg mixture over the salsa; sprinkle with 2 tbsp of cheese and ½ tbsp of cilantro. Roll up; and place, seam side down in a lightly greased 13x9 baking dish.
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Make the cheese sauce: melt the butter in a heavy saucepan over medium-low; whisk in the flour until smooth. Cook, whisking constantly, 1 minute. Increase the heat to medium. Gradually whisk in the milk; cook whisking constantly, until thickened, 5 minutes. Remove from the heat; whisk in the cheese, chiles and salt.
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Pour the cheese sauce over the rolled tortillas and sprinkle remaining cheese on top. Bake at 350 F until the sauce is bubbly, 30 minutes. Serve with tomatoes, cilantro and avocado, if desired.