Add the chorizo to a small saucepan over medium heat and cook, breaking it up as it renders, until fully browned and slightly crispy with the fat released.
Stir in the tomato paste and oregano and cook for 2 to 3 minutes until the tomato paste darkens in color and the mixture becomes fragrant and concentrated.
Pour in the rice wine vinegar and let it simmer for 1 to 2 minutes to bring the vinaigrette together without fully reducing. Remove from heat, season with black pepper, and adjust salt only if needed.
Shuck the oysters and place them on the half shell, keeping the natural liquor intact. Arrange them so they sit level to prevent spilling.
Spoon a small amount of the warm chorizo vinaigrette over each oyster, distributing evenly.
Place the oysters on a hot grill over high heat or in a 400°F oven. Drizzle lightly with olive oil and cook for 5 to 7 minutes until the oysters are just set and the sauce is bubbling.
Remove from heat and immediately grate Parmigiano Reggiano over each oyster. Let it melt slightly, then serve hot.