Heat oven to 425 F. Line a rimmed sheet pan with parchment paper.
In a large bowl, toss together the onion, pepper, tomatoes, mushrooms, garlic, thyme, rosemary, olive oil and a pinch each of salt and black pepper. Transfer the vegetables to the sheet pan.
Roast until the vegetables begin to caramelize, 25 to 30 minutes. Stir in the spinach and red pepper flakes and roast for an additional 5 minutes. Set aside.
Reduce the oven temperature to 350 F.
In a large bowl, whisk together the eggs, cream, feta, goat cheese, 1 teaspoon salt and black pepper to taste. Set aside.
Unwrap and unroll the phyllo dough and cover it with a damp cloth. Place 1 sheet of phyllo dough in a 9x13-inch pan, pressing it into the bottom and up one side. Quickly and lightly, brush the phyllo with the melted butter.
Repeat with the remaining phyllo sheets, alternating the placement of the dough, so all sides of the pan are evenly covered to form a crust.
Set aside 1 cup of the roasted vegetables. Scatter the remainder in the pan. Pour the egg mixture into the pan and scatter the reserved vegetables on top.
Bake for 40 to 50 minutes until the filling is set and the crust is golden brown and crisp. Let cool for 10 minutes. Garnish with chopped parsley and basil. Enjoy!