Heat a large skillet over high heat. Add the pasta and cook, stirring, until toasted, 5 to 6 minutes. Remove from the heat, then add the tomatoes, chicken broth, 1 teaspoon salt and a few grinds of pepper; set aside.
Season the chicken generously with salt and pepper. Heat 2 tablespoons olive oil in another large skillet over medium-high heat. Add the chicken and cook until golden brown on both sides, about 5 minutes; remove to a plate. Add the remaining 1 tablespoon olive oil to the skillet and reduce the heat to medium. Add the onion, poblano, garlic and cumin; cook, scraping up any browned bits, until the vegetables are tender, 6 to 8 minutes. Season with salt and pepper.
Add the pasta mixture to the skillet with the vegetables and bring to a boil. Nestle the chicken in the pasta; cover, reduce the heat to low and cook until the chicken is cooked through, about 12 minutes. Remove the chicken to a cutting board to rest, then continue to simmer the mixture until most of the liquid is absorbed, about 3 minutes.
Slice the chicken. Divide the pasta among plates and top with the chicken, parsley and queso fresco.