Pierce each potato twice (once on top and once on bottom). Microwave 5 - 8 minutes until nearly soft, set aside to cool slightly then cut into quarters while chicken is cooking.
Pound thicker parts of chicken to an even thickness. Season top of chicken with salt and pepper and sprinkle 1 tsp of the rosemary and 1 tsp of the thyme evenly over chicken.
Melt 1 Tbsp butter in a large skillet over medium-high heat and add olive oil. Add chicken to skillet seasoned side down, cover with a splatter screen if you have one.
Allow to cook until bottom is golden brown, about 6 minutes, then sprinkle tops with salt and pepper and 1 tsp rosemary and 1 tsp thyme.
Flip chicken to opposite side and allow to cook until center registers 165 degrees, about 6 minutes longer. Transfer chicken to a sheet of foil, spritz with juice of half of the lemon (1 1/2 Tbsp), wrap to keep warm.
Melt remaining 1 Tbsp butter in same skillet over medium-high heat and add olive oil. Add garlic and saute until lightly golden, about 30 seconds then add potatoes and remaining 1 tsp rosemary.
Cook tossing occasionally for 3 minutes then add green beans. Season veggies with salt and pepper to taste and cook until veggies are tender, about 4 minutes longer.
Spritz with juice of remaining lemon half (1 1/2 Tbsp juice), return chicken to skillet. Sprinkle with a little more rosemary and thyme and serve immediately.