Pour oil into a 5- to 6-quart pan over medium-high heat; when hot, add onion, bell pepper, and garlic and stir often until limp, 5 to 7 minutes.
Rinse chicken. Add broth, tomatoes, enchilada sauce, and chicken to pan; bring to a boil. Reduce heat, cover, and simmer, stirring occasionally, until chicken is no longer pink at the bone (cut to test), about 40 minutes.
Meanwhile, in a bowl, mix flour, cornmeal, baking powder, salt, and jalapeños. In a small bowl, whisk butter into milk; stir into flour mixture until well blended. Drop batter in tablespoon portions into simmering chicken mixture; cover and simmer gently until dumplings are cooked all the way through (cut to test), 10 to 12 minutes.
Ladle chicken, dumplings, and sauce equally into wide, shallow bowls and serve immediately.