Heat the oil in a large skillet with a lid over medium-high heat. Crumble in the breakfast sausage and add the onion. Cook, stirring occasionally, until the onion is tender, and the sausage is no longer pink, about 5 minutes. Add the cauliflower, bell pepper, garlic powder, 1 teaspoon of salt, and the pepper. Stir to combine, lower the heat to medium, and cover the pan. Let cook until the cauliflower and bell pepper are tender, about 8 minutes.

While the hash is cooking, prepare the poached eggs. Pour about 1½ - 2 inches of water into a deep skillet or pot. Add 2 teaspoons of salt and the vinegar. Bring to a boil then reduce temperature to a simmer. Carefully crack the eggs into the poaching liquid. Gently spoon the poaching water over the eggs while they cook for about 3 minutes. The eggs are done when the whites are opaque, and they feel firm to the touch. Gently remove from the poaching water and lay on a double thickness of paper towels to drain.

When the cauliflower is tender, remove the lid from the pot, raise the temperature to high, press the hash down into an even layer in the pan and let cook, undisturbed, until browned on the bottom, about 3 minutes. Flip the hash in sections, press down into an even layer, and let cook, undisturbed, until the bottom is browned, about 3 minutes.

Serve the hash topped with the poached eggs.