Wash and dry all produce. Adjust rack to upper position and preheat oven to 425 degrees. Place 2 TBSP butter in a small bowl and set aside. Halve and peel shallot, then mince until you have 2 TBSP. Trim woody bottom ends from asparagus. Cut potatoes into 1/2-inch cubes. Pick parsley leaves from stems; discard stems. Finely chop leaves.

Place potatoes in a medium pot with enough salted water to cover by 1 inch. Bring to a boil and cook until tender, about 12 minutes. Meanwhile, place milk and 1 thyme sprig in a small pot. Bring to a simmer, then remove from heat. Once potatoes are cooked, drain thoroughly and return to pot. Mash in pot with a potato masher until smooth.

Melt another 1 TBSP butter in a medium pan over medium heat. Add half the minced shallot. Cook, tossing, until just softened, 1-2 minutes. Pour shallot and melted butter into small bowl with reserved room-temperature butter. Add truffle zest and a pinch of parsley. Mash together with a fork. Set aside in freezer to chill for 10 minutes

Heat a drizzle of oil in pan used for shallot over medium-high heat. Pat steak dry with a paper towel. Season with salt and pepper. Add to pan and cook to desired doneness, 3-6 minutes per side. Remove from pan and set aside to rest. Meanwhile, toss asparagus with a drizzle of oil on a baking sheet. Season with salt and pepper. Roast in oven until just tender, about 7 minutes.

Wipe out pan after removing steak, then reduce heat under pan to low and heat another drizzle of oil in it. Add remaining shallot. Cook, tossing, until starting to soften, 1-2 minutes. Stir in demi-glace and 3 TBSP water. Let simmer until slightly thickened, about 2 minutes. Season with salt and pepper. Meanwhile, slice steak against the grain. Discard thyme sprig from pot with milk.

Stir enough milk into potatoes to create a creamy consistency (start with 1/4 cup and add more from there). Season with salt and pepper. Divide between plates along with asparagus. Arrange steak on top of potatoes. Spoon sauce over steak, then dollop with truffle butter. Garnish with remaining parsley.