Preheat the oven to 400°F.
Peel the sweet potatoes and slice them into ½-inch cubes. You should have about 4 cups. Combine the sweet potatoes with 1 tablespoon cooking oil and ¾ teaspoon salt. Spread out on a rimmed baking sheet. Roast, flipping the potatoes with a spatula halfway through cooking, until the potatoes are tender but still hold their shape, about 20 minutes.
While the potatoes roast, quarter the mushrooms and chop the bacon. Drain the kimchi and roughly chop it.
Heat your largest heavy-bottomed skillet over medium heat. Add the chopped bacon and cook, stirring occasionally, until crisp, 8 to 10 minutes. Use a slotted spoon to transfer the bacon to a paper towel-lined plate to drain.
There should be an even layer of bacon fat in the bottom of your pan. If it looks excessive, like it might slosh around when you add the mushrooms, carefully drain some out and return the pan to medium heat. Add the mushrooms and cook them, stirring occasionally, until very tender, 6 to 8 minutes. Stir in the kimchi and ¼ teaspoon of salt.
Stir together the sweet potatoes, mushrooms, and kimchi. Set aside, covered to keep warm.
Heat a nonstick pan or the pan you used to cook the hash with 1 tablespoon of cooking oil over medium heat. When the oil is hot, crack the eggs into the oil. Cook the eggs on one side for sunny-side up (if you have a lid that will fit over the top, cover the pan briefly to help the egg whites set) or flip and cook on the other side for over-easy.
Divide the hash between serving plates and top each serving with bacon and eggs. Add some chopped green onions on top, if using. Season the eggs with some extra salt and black pepper and serve!