Stir in the vinegar, transfer to a bowl, and let cool.
Stir in the garlic, lemon juice, and Tabasco.
Soften the remaining 4 tablespoons of butter and combine thoroughly with the leek mixture. Season with salt.
Light a grill. Place a dollop of leek butter on each oyster. Grill the oysters over high heat until just cooked through and the butter is melted, about 1 minute. Serve right away.