Wash, halve, deseed and slice delicata squash into ¼ inch slices. Place on a pan and toss with 1 tablespoon of olive oil and ½ teaspoon of salt. Roast at 350° F for 15 minutes, flipping the squash halfway through the cooking process for even browning. When tender, pull squash from the oven and set aside to cool. Turn up the oven to 375°F. You will use the oven to finish off the frittata in a moment.
In a 10 inch cast iron skillet, or oven-safe pan, melt one tablespoon of butter. Add chopped onion and sauté until soft and translucent.
Chop about one cup of the roasted squash and add it to the onions, along with two cloves of minced garlic. Saute for one minute.
In a mixing bowl, whisk together eggs, milk, salt, pepper and thyme. Pour eggs into the skillet, reduce heat to medium-low, cover the pan and cook for 18-20 minutes or until the frittata is almost (but not entirely) set.
Uncover the frittata and layer the remaining squash on top, followed by the shredded parmesan. Place in the oven and bake for an additional 5-8 minutes until the frittata has set completely and the cheese has melted.
Let the frittata cool for about 5 minutes before slicing and serving. Garnish with fresh thyme, if desired.
