Preheat the oven to 425 degrees F. Place the squash halves in a large microwave-safe bowl with 1/4 cup water; cover loosely with plastic wrap. Microwave until fork-tender, about 15 minutes.
Meanwhile, heat the olive oil in a medium saucepan over medium-high heat. Add the onion and poblano and cook, stirring, until slightly softened, about 2 minutes. Add the garlic, cumin, 1/2 teaspoon salt and a few grinds of pepper; cook, stirring occasionally, until the vegetables soften, about 3 minutes. Add the enchilada sauce and chicken. Bring to a simmer and cook until the sauce slightly thickens, about 5 minutes. Stir in the cilantro and season with salt and pepper.
Place the acorn squash cut-sides up in a 9-by-13-inch baking dish. Divide the chicken filling evenly among the squash halves and top with the cheese. Bake until the cheese is lightly golden and melted, 10 to 12 minutes. Top with sour cream and more cilantro.