Step 1 Prepare the Brussels sprouts: Trim the dried bases and pull away any loose or discolored outer leaves. Stand the sprout upright on the cutting board and slice it as thin as possible into coins. Aim for pieces as thin as a quarter.
Step 2 In a small lidded jar, combine the vinegar and minced shallot, and set aside for 10 minutes. Add the mustard and walnut oil and shake well to make a smooth emulsion. Taste and season with salt as needed.
Step 3Heat the olive oil over high heat in a large nonstick skillet. When the oil is very hot but not smoking, add the Brussels sprouts, sprinkle with one-half teaspoon salt and cook, tossing, until the sprouts start to wilt, 2 to 3 minutes. The texture should be chewy crisp rather than simply crisp.
Step 4Immediately pour over the dressing and remove from the heat. Continue tossing to evenly coat the sprouts with the dressing (they will only be very lightly coated). Stir in the walnuts and serve either warm or at room temperature.