Bring a large pot of lightly salted water to a boil. Cook spaghetti in the boiling water, stirring occasionally, until tender yet firm to the bite, about 12 minutes.

Heat 2 tablespoons oil and 1 tablespoon butter in a cast iron skillet over medium heat. Add chicken tenders. Cook, turning occasionally, until chicken is no longer pink at the center and juices run clear, about 10 minutes. Remove from skillet.

Add 1 tablespoon olive oil and 2 tablespoons butter to the cast iron skillet. Add fresh garlic; cook and stir until fragrant, about 1 minute.

Add broth; bring to a simmer. Cook until liquid is reduced by half, about 3 minutes.