For the Brussels sprouts: Preheat the oven to 425℉.
In a large bowl, toss the Brussels sprouts with 3 tablespoons olive oil to evenly coat, followed by the smoked paprika, salt, and pepper. (Tip: For most even coating, stir the seasonings together in a small bowl and then sprinkle with your fingers, tossing the sprouts with a large spoon as you go.) Transfer to a rimmed baking sheet (I don’t grease, as the oil on the sprouts prevents sticking) and arrange in a single layer with the cut sides facing down.
Roast the sprouts for 15-20 minutes or until golden brown on the bottom and tender. Remove from the oven and switch the oven to the broil setting.
For the chorizo: Have the garlic ready, and while the Brussels sprouts are roasting, heat the remaining 1 tablespoon olive oil in a medium (9- or 10-inch) skillet over medium heat. Sauté the chorizo, breaking it up as you go, until nicely browned. Add the minced garlic in the final minute, sauteing until you can smell it, but it hasn’t browned. Remove the pan from the heat and let sit until the sprouts are ready.
When the sprouts are done, toss them with the vinegar, and then push them together so there’s an even layer without much space between. Spoon the chorizo evenly over top, and then sprinkle with the Manchego. Broil for a minute or two, watching very closely so as not to burn, or until the cheese melts and is golden in spots.