Heat 1 tablespoon olive oil in a large skillet over medium heat. Add sweet potato and cook for 5–6 minutes until lightly browned. Add 2 tablespoons water, cover, and cook until tender.
Add onion, poblano pepper, salt, and pepper. Cook covered for 3–4 minutes until soft. Stir in chili powder and cook for 1 minute. Add black beans, mix, and remove from heat.
In a bowl, whisk eggs with salt and pepper. Heat remaining oil in a nonstick skillet and cook eggs, stirring gently, until softly scrambled.
Assemble Sweet Potato Breakfast Burritos by dividing the filling, eggs, cheese, and kale among tortillas. Fold sides and roll tightly.
Serve immediately or wrap individually for storage.