Heat the oil in a large skillet over medium-high heat. Season the chicken with ½ teaspoon salt and ¼ teaspoon pepper.
Cook the chicken until golden, 3 to 4 minutes per side. Transfer to a plate.
Add the broth, bulgur, garlic, ¾ teaspoon salt, and ¼ teaspoon pepper to the skillet and bring to a boil.
Nestle the chicken in the bulgur, reduce heat to low and cover. Simmer until bulgur is tender and chicken is cooked through, 15 to 17 minutes.
Transfer the chicken to individual plates. Add the spinach and basil to the bulgur in the skillet, stir until slightly wilted, and serve.