Place potatoes in a large sauce pot and cover with cold water. Cover and bring to a boil over high heat. Remove cover and boil until very tender, 10-15 minutes. Drain and return potatoes to warm pot and allow to steam and dry for 5 minutes.

Run potatoes through a ricer, food mill, or tamis into a large mixing bowl. Allow to cool for 5 minutes, then add butter, 2 tablespoons heavy cream, and yolks. Use a rubber spatula to mix gently just until homogenous. If mixture is dry or lumpy, add 1 to 2 additional tablespoons of heavy cream until a smooth, pipeable mixture forms. Season with salt and pepper, then set aside to cool slightly.

Line a sheet pan with parchment paper.

Transfer potato mixture to a large piping bag fitted with a large star tip. Pipe swirls of potato onto prepared baking sheet, approximately 3 inches in diameter and 3 inches high. Duchess potatoes should not spread while baking, so can be piped 1-2 inches apart.

If time allows, place potatoes in refrigerator to chill for 30 minutes to 1 hour; this helps piped edges hold their shape.

Preheat oven to 400°F, then remove potatoes from refrigerator.

Brush potatoes with whisked egg wash, then bake for 20-25 minutes, or until edges are deeply golden and crisped. During the last 10 minutes of the baking time, start the chicken.

Remove chicken from refrigerator while oven preheats.

Place chicken on a large plate or small pan and drizzle both sides of chicken breasts with vegetable oil, brushing or rubbing to coat evenly, then salt both sides generously. Set side.

Heat a large, oven safe skillet over medium high heat, then add enough vegetable oil to just coat the surface of the pan evenly. Heat until oil is shimmering, 1 minute or so, then place chicken breasts in, skin side down, pressing lightly to ensure even contact with surface of the pan. Cook until skin is crisp and deep, golden brown, about 5 minutes. Use tongs to flip chicken to tenderloin side, then place skillet in oven and cook for 5-7 minutes, or until chicken reaches an internal temperature of 160°F. (This should time so that you are removing the duchess potatoes from the oven as you put the chicken in the oven.)

While chicken is in the oven, start the asparagus.

In a large skillet over medium heat, add butter and swirl to melt and coat pan. Add trimmed asparagus, salt and pepper, and cook, tossing occasionally, for 5-6 minutes, or until bright green and very slightly softened. Add garlic and toss, cooking until fragrant, about 1 minute more. Continue cooking until asparagus is tender-crisp, another minute or two, then remove from heat, cover and set aside.

Once chicken is 160°F, remove from oven and place chicken on a cooling rack to rest and finish carry over cooking while you make the pan sauce.

If needed, pour off excess oil from pan so that only about a tablespoon remains. Heat pan over medium high, then add shallots and stir, cooking until fragrant, 30 seconds to a minute. Add lemon juice and scrape up fond from skillet. Cook until lemon juice is reduced by half, 1-2 minutes. Add mustard and whisk well to incorporate, then add chicken stock and cook on high heat until reduced by 75% and thickened (it should coat the back of a spoon).

Remove from heat and add butter, 1 tablespoon at a time, stirring/whisking/swirling to combine. Sauce should be thick and glossy. Taste and adjust seasoning if needed.

Warm both serving plates in a low oven or microwave before serving. Pan sauce is best served beneath the chicken to retain the crispy skin.