Preheat oven to 400°F (200°C) and line a large baking sheet with parchment paper.
Dice the yellow potato into small 1/2-inch cubes and cut the broccoli into bite-sized florets.
Cut the chicken breast into small pieces, ensuring they are roughly the same size as the potato cubes for even cooking.
In a large mixing bowl, combine the potato, broccoli, and chicken with the olive oil, sea salt, black pepper, garlic powder, and smoked paprika.
Toss the ingredients thoroughly until everything is evenly coated in the oil and spices.
Spread the mixture onto the prepared baking sheet in a single layer, ensuring no overcrowding to allow for maximum crispiness.
Roast for 22-25 minutes, tossing the hash halfway through, until the potatoes are golden-brown and the chicken is fully cooked.
While the hash is roasting, prepare the egg by poaching it in simmering water until the white is set but the yolk remains jammy.
Divide the roasted hash into a bowl and top with the poached egg before serving immediately.