It's high time the humble pork chop got dressed up and treated like the belle of the ball. We're giving this bone-in pork chop the steakhouse treatment with the classic trimmings usually reserved for hoity-toity cuts of beef. Duchess potatoes are like mashed potatoes taken to further perfection by baking them until they're golden brown and delicious. Châteaubriand is an old-school French sauce made with white wine, shallots, and demi-glace that's delicious on just about anything, but it's sheer perfection with this tender chop.
Peel potatoes and cut into 1” cubes. Bring a small pot with cubed potatoes, 1 tsp. salt, and enough water to cover to a boil. Reduce to simmer and cook until tender, 14-16 minutes. While potatoes boil, peel and mince shallot. Stem tarragon and coarsely chop. Peel carrot, trim, and cut into sticks 3" long and ½" thick. Rinse pork chops, pat dry, and season both sides with ½ tsp. salt and a pinch of pepper.
Finish the Potatoes
Once cooked, drain potatoes in colander and return to pot. Add cream, ⅓ the butter (reserving remaining for sauce), and liquid egg to pot and mash until smooth. Season with a pinch of salt and pepper. Transfer to prepared casserole dish or two oven-safe ramekins. Use a fork to pull sides of potatoes into small peaks, forming decorative ridges. Bake until ridges on potatoes turn golden brown, 20-25 minutes. Remove from oven and set aside. While potatoes bake, roast carrots.
Roast the Carrots
Place carrots on prepared baking sheet and toss with 1 tsp. olive oil, ¼ tsp. salt, and a pinch of pepper. Spread into a single layer and roast until browned and tender, 12-15 minutes. While carrots roast, cook pork chops.
Cook the Pork
Place a medium non-stick pan over medium-high heat. Add 1 tsp. olive oil and pork chops to hot pan. Cook until browned, 3-5 minutes. Flip, reduce heat to medium, and cook until pork chops are firm and reach a minimum internal temperature of 145 degrees, 4-8 minutes. Remove from pan and rest 3 minutes. Reserve pan; no need to wipe clean.
Make the Sauce
Return pan used to cook pork to medium-high heat. Add shallots and ½ tsp. olive oil. Cook until aromatic, 1 minute. Add white cooking wine and demi-glace. Bring to a boil and reduce by half, 2-3 minutes. Remove from burner and swirl in remaining butter and tarragon.
Plate the Dish
If carrots are cool, return them to oven to reheat, 1-2 minutes. Set duchess potatoes onto plate and set roasted carrots next to them. Place pork chop in front of vegetables and spoon sauce in front of chop.