Heat 1 tablespoon of olive oil in a large skillet over medium-high heat. Once hot, add the diced butternut squash and spread in an even layer. Let it cook undisturbed for 3-4 minutes to develop a golden crust. Stir and continue cooking for another 5-7 minutes until the squash begins to soften but still holds its shape. Transfer to a plate and set aside.
In the same skillet, add the ground sausage, breaking it apart with a wooden spoon as it cooks. Brown for about 5-7 minutes until no longer pink. If there's excess fat, drain all but about 1 tablespoon from the pan.
Add the diced onion to the sausage and cook for about 3 minutes until translucent, stirring occasionally.
Stir in the diced apples and garlic and cook for another 2-3 minutes. The apples should soften slightly but maintain some texture.
Return the butternut squash to the skillet along with the sage, thyme, cinnamon, nutmeg, salt, and pepper. Stir gently to combine all ingredients.
Pour in the chicken broth or apple cider and scrape up any browned bits from the bottom of the pan. These bits are flavor gold!
Cook for another 5 minutes until most of the liquid has evaporated and the squash is fork-tender but not mushy.
If using greens, add them now and stir until just wilted, about 1-2 minutes.
Taste and adjust seasonings as needed. Remove from heat.
If serving with eggs, make four wells in the hash using the back of a spoon. Add the remaining tablespoon of olive oil to the wells and crack an egg into each. Cover the skillet and cook over medium-low heat until the eggs are set to your liking, about 3-5 minutes for runny yolks.
Garnish with pepitas or chopped pecans for a delightful crunch before serving.