Trim ends off green beans. Cut mushrooms into ¼" slices. Stem and mince rosemary. Mince garlic. Pat chicken breasts dry, and season both sides with a pinch of salt and pepper.
Start the Green Beans and Chicken
Place green beans on prepared baking sheet and toss with garlic, 2 tsp. olive oil, ¼ tsp. salt, and ¼ tsp. pepper. Massage oil and seasoning into green beans. Spread into a single layer on one half of baking sheet (some overlap is ok). Roast in hot oven, 5 minutes. While green beans roast, heat 1 tsp. olive oil in a medium non-stick pan over medium heat. Add chicken to hot pan. Cook undisturbed until browned, 4-5 minutes. Chicken and green beans will finish cooking in a later step.
Finish the Green Beans and Chicken
Transfer chicken, seared side up, to empty half of baking sheet. Roast until green beans are tender and chicken reaches a minimum internal temperature of 165 degrees, 10-12 minutes. Reserve pan; no need to wipe clean. While chicken and green beans roast, make mushroom gravy.
Make the Mushroom Gravy
Return pan used to sear chicken to medium heat and add 1 tsp. olive oil. Add mushrooms and rosemary (reserving a pinch for garnish) to hot pan. Stir occasionally until browned, 2-3 minutes. Add sherry and stir occasionally until slightly thickened, 3-4 minutes. Add cream and chicken base and stir occasionally until liquid reaches the consistency of a light gravy, 1-2 minutes.
Finish the Dish
pour sauce on chicken and garnishing chicken and sauce with reserved rosemary