Combine the vegetables and garlic on a sheet pan with the olive oil and spices. Bake for 10 minutes, until beginning to crisp.
Remove baking sheet from oven and make 6 indentations. Crack eggs individually into a small bowl and carefully slide 1 egg into each indentation. Sprinkle eggs with a bit of sea salt and cracked black pepper. Return sheet pan to oven and cook for 5-7 more minutes, until the whites are set and yolks are still a bit runny. If you prefer your yolks cooked through, leave in the oven for an additional 2 minutes.
Sprinkle eggs with dried oregano and serve.
*if you need to make this in advance, undercook the eggs slightly. Then reheat the hash and eggs in the oven at 425 degrees F. for only 2-3 minutes.