Preheat oven to 375°. Line a large rimmed baking sheet with foil.
In a medium bowl, combine squash, 1 tablespoon oil, 3⁄4 teaspoon salt, and thyme. Arrange in a single layer on prepared pan.
Bake until lightly browned and tender, about 20 minutes. Discard thyme. Remove squash and foil; set aside. Let pan cool.
Increase oven temperature to 425°. Lightly spray same pan with cooking spray.
On a lightly floured surface, roll dough into a 12-inch circle. Transfer to prepared pan. Brush with remaining 1 tablespoon oil. Sprinkle cheese on top in an even layer, leaving a 1⁄2-inch border around edges. Arrange onion, sausage, and squash on top.
Bake until crust is lightly golden brown and cheese is melted, 10 to 15 minutes. Using a spoon or small measuring cup, carefully press to create 4 indentations in top of pizza. Crack 1 egg into each indentation, and sprinkle evenly with remaining 1⁄4 teaspoon salt. Bake until egg whites are set and yolks are cooked to desired degree of doneness, 8 to 12 minutes more. Garnish with red pepper, if desired.
Recipe by Paula Deen Magazine at https://pauladeenmagazine.com/butternut-squash-breakfast-pizza/