Preheat the oven to 400F.
In a large mixing bowl or on a sheet tray, coat the lamb chops in vegetable oil and season the outsides with some Kosher salt.
Add the chopped rosemary to the exterior of the chops, trying to distribute the rosemary relatively evenly over the surface of the meat.
Allow the lamb chops to sit at room temperature for 30 minutes.
After the lamb chops have rested at room temperature for 30 minutes, place them on a sheet tray to roast in the oven at 400F.
Roast the lamb chops for roughly 20 minutes, or until the meat is cooked to your liking.
*Thin lamb chops are going to cook a lot faster than double-rib chops.
While the lamb is roasting, boil your Fingerling potatoes in salted water until they are tender, and set them aside.
While the potatoes are boiling, boil the green beans in a separate small soup pot until right before they are fully tender.
When the green beans are barely done cooking and are just about to become fully tender, drain the water and set the green beans aside.
*The green beans will continue to cook slightly at room temperature after they have been removed from the boiling water.
When the lamb chops are done roasting, carefully remove them from the sheet tray and set them aside on a plate to rest.
Deglaze the sheet tray with 3 Tbsp of red wine, using a wooden spatula to loosen the browned bits and lamb drippings from the bottom of the sheet tray.
Pour the liquid and drippings from your lamb roasting sheet tray into a small sauce pot.
Add ½ cup of vegetable or chicken stock to the small sauce pot, and bring contents to a simmer.
In a small bowl, whisk together 1 Tbsp corn starch with 1 Tbsp cold water.
*This mixture of corn starch and cold water will be added to your simmering sauce and will be used as a thickener.
When the stock is simmering, whisk in the cornstarch and water mixture.
When you have achieved the desired sauce thickness that you desire, turn off the heat and stir in 4 Tbsp of cubed salted butter.
Continue to stir until the butter is emulsified into the sauce.
Add a teaspoon or two of sherry vinegar, and taste to see if you would like to add more vinegar.
Season with kosher salt and freshly ground black pepper to taste.
When the lamb chops are cooked and your sauce is finished, you are ready to serve.
Plate the lamb chops with a spoonful or two of the sauce.
Add the boiled Fingerling potatoes and green beans, topped with a bit of softened butter.
Garnish each plate with some finely-sliced mint.