Heat the oil a large non stick pan on medium heat. While heating, season the chicken breasts on both sides with salt and pepper as desired. Add chicken to the hot pan and cook to nicely brown each side. Remove chicken from pan and set aside.

Same pan, add the chicken broth and broccoli and bring to a boil. Reduce heat to a simmer and cover, cooking the broccoli until just tender, about 3-5 minutes. Remove the broccoli to a bowl with a slotted spoon, keeping as much as the broth as possible in the pan. Cover the bowl to keep the broccoli warm. Remove pan from heat.

In a small bowl, whisk together the milk and flour until smooth then whisk in the dill.

Place the same pan back on medium heat and stir the milk mixture into the broth and bring to a boil while stirring constantly. Stir until thickened, about 1-2 minutes.

Add the chicken back to the pan and reduce heat to a simmer, cover, and allow the chicken to simmer in the sauce for 10-13 minutes or until cooked through. Sacrifice one piece and cut in half to check, do not over cook the chicken or it will be dry.

Serve the chicken with a side of broccoli and spoon some of the sauce over the chicken.