Heat half the olive oil in a skillet over medium-high heat. Add the onions and let cook for 7-10 minutes until softened and just beginning to caramelize. Season with salt and pepper. Add the hash browns and stir to combine with the onions. Press down into the pan and let them crisp up on the bottom, cooking them for 3-4 minutes without moving them. Stir and repeat to get nice and crispy potatoes. Remove from the pan and set aside.
Add the remaining olive oil to the skillet with the garlic and red pepper flakes. Cook for 1 minute. Add the zucchini, red pepper, broccoli, and mushrooms. Cook for 7-10 minutes until tender-crisp, adding couple of tablespoons of water to the pan if the vegetables begin to burn. During this stage, cook the veggies to your desired tenderness. I prefer them to have some crunch still but you can cook them longer as well.
Add the hash brown potatoes and onions back to the skillet and toss everything together. Cook for 1-2 minutes and season with salt and pepper. Serve with feta cheese on top. Many people eat this as a breakfast hash with eggs and toast but it also works great for lunch or dinner.