Preheat your oven to 375°F (190°C).
Place the poblano peppers on a baking sheet and roast them in the oven for about 15 minutes, or until the skin is charred and blistered. Remove from the oven and let them cool. Once cooled, peel off the skin, remove the seeds, and dice the peppers.
In a large skillet, heat the olive oil over medium heat. Add the diced onion and minced garlic, and sauté until they become translucent and fragrant.
Season the chicken breasts with cumin, chili powder, salt, and pepper. Add them to the skillet and cook until they are no longer pink in the center. Remove the chicken from the skillet and shred it using two forks.
In a mixing bowl, combine the shredded chicken, diced poblano peppers, half of the shredded Monterey Jack cheese, half of the shredded cheddar cheese, and half of the sour cream. Mix well.
Pour half of the chicken broth into a baking dish.
Warm the flour tortillas in the microwave for about 30 seconds to make them more pliable. Spoon the chicken mixture onto each tortilla, roll them up, and place them seam-side down in the baking dish.
Pour the remaining chicken broth over the enchiladas, and top them with the remaining shredded cheeses.
Bake in the preheated oven for about 20 minutes, or until the cheese is melted and bubbly.
Serve the enchiladas hot, topped with dollops of sour cream and any desired garnishes such as chopped cilantro or sliced avocado.