Cut potatoes into ¼" dice. Stem and mince mint and parsley. Zest lime, halve, and juice. Trim ends off green beans.
Make the Mashed Potatoes
Bring a small pot with potatoes and enough salted water to cover to a boil over high heat. Reduce to a simmer and cook until tender, 15-18 minutes. Drain potatoes in a colander and return to pot. Add buttermilk and butter and mash until desired consistency is reached. Season with a pinch of salt and pepper. While potatoes cook, roast lamb.
Sear and Roast the Lamb
Pat lamb chops dry, and season both sides with ½ tsp. salt and ¼ tsp. pepper. Place a large pan over medium-high heat. Add 2 tsp. olive oil and lamb chops to hot pan. Sear until well- browned, 3-5 minutes. Transfer chops to prepared baking sheet, seared side up, and roast until chops reach a minimum internal temperature of 145 degrees, 8-10 minutes. Rest lamb 5 minutes. Pour off all but 2 tsp. oil out of pan and reserve remaining in pan. While lamb roasts, make chimichurri.
Make the Chimichurri
In a mixing bowl, combine parsley, mint, half the lime zest, 2 tsp. lime juice, 1 Tbsp. olive oil, red pepper flakes (to taste), and a pinch of salt and pepper.
Cook the Green Beans
Once lamb is resting, return pan used to sear lamb to medium-high heat. Add green beans to hot pan and cook until lightly browned, 2-3 minutes. Add ¼ cup water and a pinch of salt and pepper and cook until green beans are tender, 2-4 minutes. Remove from burner.
Finish the Dish
Place green beans on a plate. Place potatoes and lamb chops next to green beans and top lamb with chimichurri.