Unroll the pie crusts and press them into a quarter sheet pan. The crusts will overlap in the middle. Press the seam together. Then fold the excess dough around the edges to be even with the sides of the pan. Chill while you prepare the filling.
Place the thawed broccoli in a colander to drain excess moisture, squeezing with a paper towel if necessary. (Note 1) Set aside.
In a small skillet set over medium heat, add the butter and the diced onions and saute until softened and translucent, approx. 5 minutes. Season with salt and pepper and stir in the fresh thyme. (Note 2)
In a mixing bowl, combine 1 cup of the cheddar cheese with the parmesan cheese, broccoli, and onion mixture. Stir to combine.
Spoon the filling into the pie crust-lined pan and spread evenly. Top with remaining cheese.
Whisk together the egg with 1 teaspoon water. Brush the edges of the pie crust. Sprinkle with everything bagel seasoning, sesame seeds, or pepper, if desired.
Bake for 25 minutes until the crust is golden brown and the cheese is bubbly and lightly brown. Let cool for 5 minutes. Cut into squares and serve.
Notes
If using fresh broccoli, blanch the broccoli before combining it with the rest of the filling ingredients. Add the broccoli to boiling water for 30-60 seconds before transferring it to a bowl filled with ice water. Drain well to remove any excess moisture. Chop.
Fresh thyme can be substituted with 1 ½ teaspoons dried thyme.