Heat oven to 400°F. Toss sweet potato cubes with 2 teaspoons oil and salt on a parchment-lined rimmed sheet pan. Roast 25–30 minutes, turning once, until fork-tender. Lightly mash about half the cubes with a fork, leaving the rest in small pieces.

Meanwhile, heat remaining teaspoon oil in a skillet over medium heat. Cook onion 5–6 minutes until softened. Add garlic, cumin, and chili powder; stir 30 seconds. Add beans, corn, and water; cook 3–4 minutes until hot. Stir in roasted sweet potato and lime juice.

Spread about 1/2 cup salsa verde across the bottom of a 9-by-13-inch baking dish. Warm tortillas in small stacks wrapped in a damp paper towel in the microwave, 20–40 seconds or until pliable. Keep covered and warm as you work.

Divide the filling among the 10 tortillas, approximately a generous 1/3 cup each. Roll gently and place seam-side down in the dish. If a little filling will not fit, tuck it between the rolls rather than forcing them closed. Spoon all remaining salsa over the tops, covering exposed edges, then sprinkle evenly with cheese.

Cover loosely with foil and bake 20 minutes. Uncover and bake another 5–10 minutes, until sauce bubbles and the centers reach 165°F. Rest 5 minutes, scatter cilantro on top, and serve 2 enchiladas per person.

Refrigerate leftovers in shallow containers within 2 hours, or 1 hour above 90°F. Use within 3–4 days. Reheat covered with a small splash of water until the centers reach 165°F.