Preheat the oven to 325 degrees.

Toss the walnuts with olive oil, salt, and black pepper, spread on a baking sheet, and toast until deeply golden, fragrant and crispy, 8-12 minutes. (If after 12 minutes the texture is still weak or waxy, leave them in longer- when properly toasted they have superbly satisfying crunch.) Let cool, and then chop or crumble each into a few smaller pieces.

Turn the oven up to 350 degrees.

Make the vinaigrette: finely mince the shallot and macerate with 1 Tbsp sherry vinegar for 10 minutes. In a medium bowl, whisk together the mustard, macerated shallot, and maple syrup or honey with a little of the sherry vinegar. Add the olive oil in a slow drizzle to emulsify, alternating with the sherry vinegar. Adjust with salt and pepper.

Toss the Brussels sprouts in a bowl with a little olive oil, salt, and black pepper to coat.

Set a large cast-iron skillet over high heat and wait until the pan is very hot to add oil, just to coat the bottom of the pan. Working in batches, place the Brussels sprouts in the pan, cut side down, and sear very hard until deeply caramelized on the flat side, then turn and get a kiss of color on the backside. Depending on the size of the sprouts, they may be fully cooked at this point (if they are very small), but they will likely need 5 to 15 minutes in the oven to finish cooking. Pull them when they are finished to your desired doneness (I love them both al dente and very tender.) Squeeze a lemon over the Brussels sprouts as they cool.

(If you are preparing a large batch, individually searing can be unrealistic. Instead, put an empty baking sheet in a hot oven for 10 minutes before adding the Brussels sprouts: it should hold enough heat to give them some color when they hit. Don’t overcrowd the sheet; leave a little negative space for water to evaporate, which helps with browning.)

To serve, drizzle with the vinaigrette and top with toasted walnuts and a few grinds of black pepper.