Preheat your oven to 400°F (200°C) and position a rack in the center of the oven.

Place the Brussels sprouts on a rimmed baking sheet. Drizzle with 3 tablespoons of olive oil and toss to coat evenly. Spread them out in a single layer and season with 1 teaspoon of salt. Roast for about 25 minutes, or until tender and golden brown.

While the Brussels sprouts are roasting, heat the vegetable oil in a 12-inch skillet over medium-high heat until shimmering. Add the thinly sliced shallots and cook, stirring frequently, until they turn golden brown, about 7 to 10 minutes. Using a slotted spoon, transfer the shallots to a paper towel-lined plate to drain. Sprinkle with a little salt and let them cool.

Discard any remaining vegetable oil from the skillet and return it to medium-high heat. Add the remaining 1 tablespoon of olive oil and butter. Once the butter has melted, add the wild mushrooms. Cook, stirring occasionally, until the mushrooms become golden brown and tender, about 5 to 8 minutes. Season with the remaining ½ teaspoon of salt.

Pour in the dry white wine and cook for 3 to 4 minutes, or until the wine reduces by half.

Add the roasted Brussels sprouts to the skillet with the mushrooms. Pour in the heavy cream and stir to combine. Continue cooking for another 5 minutes until the cream thickens and coats the Brussels sprouts. Taste and adjust the seasoning with additional salt and freshly cracked black pepper.

Serve the dish topped with the crispy fried shallots for a delightful crunch and added flavor.