The sauce served with this tender steak brings together two things we love: alliteration and awesome aromas. See, we just can't help it! But when you taste the way the nutty brown butter complements the rich red wine reduction sauce, and the crispy exterior on the roasted Brussels sprouts, you'll agree this meal kicks some assonance.
Peel potato and cut into 1 ½" dice. Trim bottoms off Brussels sprouts and halve. (Quarter if larger than ping pong balls.) Peel and mince shallot. Rinse steaks, pat dry, and season both sides with ¼ tsp. salt and ¼ tsp. pepper.
Roast the Potatoes
Place a large oven-safe pan over medium-high heat. Add 3 Tbsp. olive oil and potatoes to hot pan and cook, stirring occasionally, 1 minute. Cover, reduce heat to medium, and cook 5 minutes, stirring once halfway through. Uncover, place pan in oven, and roast, stirring halfway through, until golden brown and tender, 20-22 minutes. Remove from oven, season to taste with salt and pepper, and set aside. While potatoes roast, start Brussels sprouts.
Roast the Brussels Sprouts
Place Brussels sprouts on prepared baking sheet and toss with 2 tsp. olive oil and ¼ tsp. salt. Spread into a single layer on one half of baking sheet. Roast 6-8 minutes and remove from oven. Brussels will finish cooking in a later step. While Brussels sprouts roast, sear steaks.
Cook the Steaks
Place a medium non-stick pan over high heat. Place 2 tsp. olive oil and steaks in hot pan and sear until well-browned, 2-3 minutes per side. Transfer steaks to other half of baking sheet. Roast until steaks reach a minimum internal temperature of 145 degrees and Brussels sprouts are tender and caramelized, 8-12 minutes. Reserve pan; no need to wipe clean. While steaks roast, start sauce.
Make the Sauce
Return pan used to sear steaks to medium-low heat. Add half the butter and cook, stirring occasionally, until butter smells "nutty," turns golden, and brown flecks appear, 5-6 minutes. Add shallots and cook until aromatic, 30 seconds. Add red cooking wine, beef demi-glace, and ⅓ cup water. Raise heat to medium-high and cook until reduced by half and slightly thickened, 1-3 minutes. Remove from burner and swirl in remaining butter.
Plate the Dish
Spoon a serving of roasted potatoes on plate and arrange Brussels sprouts next to potatoes. Pour sauce on plate and serve steak over sauce.