In a skillet, melt 2 tablespoons of the butter. Add the leek and cook over moderate heat, stirring, until softened and lightly browned, 8 minutes. 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.