Line a plate with paper towels. Place a large skillet over medium-high heat. Add the vegetable oil, and when it shimmers, add the mushrooms caps in a single layer. Cook until well browned on one side, about one minute, then flip and cook the other side. When the second side is browned, remove with a slotted spatula to the paper towels, leaving as much oil behind as possible. Season with a sprinkle of Kosher salt.
With the skillet still on medium-high heat, add the brussels sprouts and toss to coat with oil, adding a bit more oil if needed. Cook, turning occasionally, until tender and many surface have dark brown spots. I like mine almost charred.
Add the smoked paprika, 1/2 teaspoon of Kosher salt and the reserved shiitakes. Toss, then taste and adjust seasoning. Transfer to a serving platter, finish with black pepper and a few grains of flaky salt and serve.