Preheat broiler with oven rack 8 in. from heat source. Toss together sweet potato, poblano, canola oil, and cumin on a large rimmed baking sheet. Broil until tender and lightly browned, 10 to 12 minutes. Remove from oven and let cool slightly for about 5 minutes.
Meanwhile, stir together salsa, broth, and cornstarch in a medium saucepan. Bring to a boil over medium-high. Reduce heat to medium to maintain a simmer. Cook, stirring often, until thickened, about 2 minutes. Spread 1/3 cup of the sauce in the bottom of an 11x7-in. baking dish coated with nonstick cooking spray. Set aside.
Coarsely mash white beans in a medium bowl. Stir in sweet potato mixture and 1/2 cup of cheese until combined. Spoon 1/4 cup of the filling along the center of each tortilla. Roll up into a cigar shape. Arrange, seam-side down, in a single layer over salsa mixture in baking dish. Top enchiladas with remaining sauce and sprinkle with remaining 1/2 cup cheese.
Broil until cheese is melted and bubbly, about 5 minutes. Sprinkle with cilantro. Serve enchiladas hot.