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1
Place room-temperature butter in a small bowl and set aside.
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2
Place chopped chorizo in the bowl of a blender or mini food processor fitted with the metal blade. Pulse, scraping down the sides as needed, until it resembles coarse, wet sand.
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3
Heat the olive oil in a large cast iron or heavy bottomed skillet set over over medium heat. Add the chorizo and cook in as close to a single layer as possible, stirring occasionally, until slightly crisp; about 4 minutes. Add minced garlic, paprika (if using) and cayenne (if using). Cook, stirring constantly, until the mixture is fragrant and the garlic softened; about 2 minutes. Remove from heat, then transfer chorizo to a paper towel-lined plate to drain. Allow mixture to come to room temperature. Place cooled chorizo in bowl with room-temperature butter, whip together with a small fork until well combined; set aside. You can make make the chorizo butter up to 3 days ahead. Keep wrapped and refrigerated until ready to use. Bring to room temperature before continuing.
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4
Prepare a gas or charcoal grill for medium-high heat. If you're using charcoal, don't wait until the coals are white and ashen, you'll want a little live fire. Using an oyster knife and a heavy glove or towel carefully shuck the oysters, keeping the shell horizontal to retain as much liquid as possible. Wipe the oyster knife clean, discard top shell, then carefully slide the oyster knife under the oyster flesh to cut through the muscle holding it to the shell. Leave the oyster in place. Repeat with remaining oysters setting them carefully on a rock salt covered serving plate as you work (if using). Once all the oysters are shucked, spoon a generous teaspoon-sized dollop of room temperature chorizo butter onto each of the raw oysters.
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5
When ready to grill place the oysters, face up, directly on the grates over lightly flaming live fire. Wait for the butter to melt and begin to sizzle, about 2 to 3 minutes. Depending on the heat of the fire, you may need to cover the grill during cooking so they cook quickly. Using tongs, carefully move the grilled oysters, retaining the liquid, to the rock salt covered serving plate (if using). Serve immediately with lime wedges on the side.
