Heat the oil in a large skillet over medium-high heat. Season the chicken with 1/2 teaspoon salt and 1/4 teaspoon pepper and cook until golden, 3 to 4 minutes per side. Transfer to a plate.
Add the broth, bulgur, garlic, 3/4 teaspoon salt, and 1/4 teaspoon pepper to the skillet and bring to a boil.
Nestle the chicken in the bulgur, reduce heat to low, cover, and simmer until the bulgur is tender and the chicken is cooked through, 15 to 17 minutes.
Transfer the chicken to plates. Add the spinach and basil to the bulgur, stir until slightly wilted, and serve.