Uncover the pan, raise the heat back to medium and cook until the excess moisture has evaporated and the butternut squash is starting to turn golden on the edges, about 5 to 10 minutes (add another little splash of olive oil if the squash starts sticking to the pan).
In a 10-inch, well-seasoned cast iron skillet or oven-safe sauté pan, warm 1 tablespoon olive oil over medium heat. Add the chopped onion and stir to coat. Cook for a few minutes, until the onions are starting to turn translucent. Add the squash and ½ teaspoon salt and stir. Cover the pan and reduce heat slightly to avoid burning the contents. Cook until the butternut is tender and cooked through, stirring occasionally, about 8 minutes.
Preheat the oven to 425 degrees Fahrenheit. In a large bowl, whisk together the eggs, milk, garlic, ¼ teaspoon salt and several twists of freshly ground black pepper. Then whisk in about half of the cheese.