Preheat the oven to 400 F.
Add the potatoes to a medium saucepan, cover with water, add 2 teaspoons of salt and bring to a boil. Reduce to a simmer, and cook for 5 minutes at a simmer until the potatoes are just barely tender but still not cooked all the way through.
In a large oven proof sauté pan, over medium heat, add the olive oil and heat. Add the sliced onions and sauté until they just begin to take on a golden color. Add the garlic and sauté another minute. Remove the pan from heat.
Drain the potatoes well and allow to sit for a minute to help the moisture evaporate. Add the potatoes to the sauté pan with the onions, tossing well, and allow to cook over medium heat until they just begin to turn golden brown. Add the roasted red peppers, thyme leaves, paprika and toss to mix.
Taste and season with additional salt and pepper, if desired.
Place the sauté pan in the oven and roast for 20 minutes until the potatoes are nicely browned.
Remove the sauté pan and carefully crack the eggs over the hash, leaving space in between each egg, sprinkle with a bit of salt and pepper and slide the sauté pan back in the over for an additional 3-5 minutes depending on how you like your eggs cooked.
Add finely chopped parsley to the pan and drizzle with additional UP olive oil and paprika if desired.