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 1/2" thick. Rinse pork chops, pat dry, and season both sides with 1/2 tsp. salt and a pinch of pepper.

Once cooked, drain potatoes in colander and return to pot. Add cream, 1/3 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.

Place carrots on prepared baking sheet and toss with 1 tsp. olive oil, 1/4 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.

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.

Return pan used to cook pork to medium-high heat. Add shallots and 1/2 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.

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.