Preheat the oven to 425 degrees F (220 degrees C). Line a baking sheet with aluminum foil.

Halve poblano lengthwise; remove seeds and membranes. Put cut sides down on the prepared baking sheet; roast until skins are charred, about 20 minutes. Wrap foil around pepper halves and let stand until cool enough to handle, 20 to 30 minutes.

Meanwhile, brown sausage in a large skillet over medium heat, stirring to break up lumps, about 5 minutes. Transfer to a plate; drain grease.

Return the skillet to medium heat; melt 1 tablespoon butter with the oil. Stir in bell peppers, sweet potato, onion, chili powder, cumin, red pepper flakes, and garlic. Cook, stirring occasionally, until vegetables are tender and starting to brown, 20 to 25 minutes.

Peel and chop the poblano; add to the skillet, along with sausage, and 1/4 teaspoon each salt and black pepper. Cook until heated through, 1 to 2 minutes. Transfer to a serving platter and keep warm.

Rinse the skillet, return to heat, and melt remaining 1 tablespoon butter. Crack eggs into the skillet; sprinkle with remaining 1/4 teaspoon each salt and black pepper. Reduce heat to low and cook until whites are completely set and yolks begin to thicken, 3 to 4 minutes.

Top hash with eggs and serve with salsa, cheese, and cilantro.