Preheat the oven to 375°F (190°C). Fill a medium pot three-fourths full with water, bring to a boil over high heat, and add 1 tablespoon of kosher salt. Plunge the halved Brussels sprouts into the boiling water and cook until they start to feel tender when pierced with a knife tip, about 8 to 10 minutes, depending on their size. Drain the sprouts well and set aside.

Heat the canola oil in a medium cast-iron or heavy frying pan over medium-high heat. Working in batches if necessary to avoid overcrowding, place the Brussels sprouts cut-side down in a single layer. Cook until the cut sides are browned and caramelized, about 6 minutes. Transfer the sprouts to a plate and reserve.

Using the same pan, add the smoked sausage slices, shallots, remaining 1 teaspoon of kosher salt, freshly cracked black pepper, and unsalted butter. Reduce the heat to medium and cook, stirring occasionally, until the shallots are softened and aromatic, about 5 minutes.

Return the caramelized Brussels sprouts to the pan and add the sherry vinegar or balsamic vinegar. Toss together to evenly coat all the ingredients. Transfer the entire pan to the preheated oven and roast until the sprouts are tender when pierced with the tip of a knife and deeply browned, about 10 minutes.

Carefully remove the pan from the oven and transfer the contents to a serving dish. Serve immediately for a warm, smoky, and tangy side dish that’s sure to be a crowd-pleaser.