Step 1: Wash the yellow squash thoroughly. Trim the ends and slice them into ¼-inch thick rounds. Place the slices in a colander, sprinkle lightly with salt, and let them sit for 15-20 minutes. This draws out excess moisture. Pat them very dry with paper towels before proceeding.
Step 2: In a large skillet, melt 1 tablespoon of olive oil or butter over medium heat. Add the diced onion and cook until softened and translucent, about 5-7 minutes. Stir in the minced garlic and cook for another minute until fragrant.
Step 3: Add the dried squash slices to the skillet with the onions and garlic. Sauté for about 5-7 minutes, stirring occasionally, until the squash begins to soften slightly but still holds its shape. Transfer the squash mixture to a large mixing bowl.
Step 4: In a separate bowl, whisk together the heavy cream, eggs, grated Parmesan cheese, dried thyme, 1/2 teaspoon salt, and 1/4 teaspoon black pepper until well combined.
Step 5: Add half of the shredded cheddar cheese and half of the shredded Monterey Jack cheese to the bowl with the squash mixture. Pour the cream and egg mixture over the squash and cheese. Gently fold everything together until well combined.
Step 6: Lightly grease a 9x13 inch baking dish. Pour the squash mixture into the prepared baking dish, spreading it evenly.
Step 7: Sprinkle the remaining cheddar and Monterey Jack cheese over the top of the squash mixture. In a small bowl, combine the Panko breadcrumbs with the 2 tablespoons of melted butter. Sprinkle this mixture evenly over the cheese layer.
Step 8: Preheat your oven to 375°F (190°C). Bake for 30-35 minutes, or until the casserole is bubbly around the edges, golden brown on top, and the cheese is fully melted and slightly caramelized. If the top starts to brown too quickly, you can loosely tent it with aluminum foil.
Step 9: Once baked, remove the casserole from the oven and let it rest for 5-10 minutes before serving. Garnish with fresh chopped parsley if desired.