Stem thyme and coarsely chop. Peel, trim, and cut carrots into ½" pieces at an angle. Place a medium oven-safe pan over medium-high heat. Add 1 tsp. olive oil and carrots to hot pan. Cook carrots, stirring often, until lightly browned, 3-5 minutes. Place pan in oven and cook until tender and golden brown, 15-18 minutes, stirring once halfway through. Remove from oven and season with half the thyme (reserve remaining for sauce), ¼ tsp. salt, and a pinch of pepper.
Sear the Steaks
Rinse steaks, pat dry and season both sides with ½ tsp. salt and ¼ tsp. pepper. Place a medium non-stick pan over medium-high heat. Add 1 tsp. olive oil and steaks to hot pan. Sear each side until browned, 2-3 minutes per side. Remove to plate. Steak will finish cooking in a later step. Reserve pan; no need to wipe clean.
Cook the Mushrooms
Slice mushrooms thinly. Return pan from cooking steak over medium-high heat. Add 2 tsp. olive oil and mushrooms to hot pan. Cook, stirring occasionally, until mushrooms are golden brown, 4-6 minutes. Season with ¼ tsp. salt and a pinch of pepper. Remove from burner. While mushrooms cook, prepare pastry.
Prepare the Puff Pastry
Use the back of small knife to draw diagonal lines on puff pastry sheet, ½" apart. Draw another set of lines in opposite direction to make a crosshatched pattern. Cut puff pastry in half and refrigerate to keep cold. Once mushrooms are cooked, place ⅓ the mushrooms (reserve remaining for sauce) on top of each steak.
Bake the Steaks
Place steaks on prepared baking sheet. Place puff pastry over the width of mushroom-covered portion of steak. Stretch dough slightly so dough makes it under the bottom edge of steak. Place on top rack in oven and bake until puff pastry is golden brown and steak reaches a minimum internal temperature of 145 degrees, 8-12 minutes. Remove from oven and allow to rest 2 minutes. While steaks bake, make sauce.
Plate the Dish
Add demi-glace and ¼ cup water to pan with remaining mushrooms. Bring to a boil over high heat and boil until reduced by half, 1-3 minutes. Remove from burner and swirl in butter and remaining thyme. Place carrots on plate and put Wellington in front of carrots. Spoon mushrooms in front of steak.