Cheesy Hash Brown Quiche

about 4 tablespoons olive oil, divided

-ounces hash browns, thawed and seasoned with salt and pepper

large eggs

shredded mozzarella cheese, divided

half-and-half (whole milk or heavy cream may be substituted)

or 3 green onions, sliced into thin rounds

kosher salt, or to taste

freshly ground black pepper, or to taste