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Preheat oven to 400 degrees. Butter a 9 x 13 inch baking dish or spray it with cooking spray.
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In a medium bowl toss the yellow squash and zucchini with ¾ teaspoon salt. Let the squash rest for 15 minutes and squeeze it dry.
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Heat the oil in a large sauté pan over medium heat. Add the onion and cook for 6-7 minutes until it is translucent but not brown.
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Add the drained squash and sauté for 7 minutes until it is just soft but not mushy. Sprinkle the squash with cornstarch and cook for 2 minutes until the cornstarch is all incorporated and the juices from the squash have turned into a creamy sauce.
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Remove the pan from the heat and set aside for 15 minutes.
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While the squash is cooling, whisk the eggs, half and half, nutmeg and the remaining ¾ teaspoon of salt.
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Spoon the cooled squash into the prepared baking dish and evenly pour in the egg mixture. Sprinkle with the cheese.
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Bake the casserole for 25 minutes until it is brown around the edges and lightly brown in the center. Allow the casserole to cool for 10 minutes and serve.