Cover; chill at least 1 hour and up to 6 hours, turning often.
Whisk broth, juice, sugar, 3 tablespoons butter, tamarind paste, peel, and ginger in large saucepan.
Boil mixture over medium-high heat until reduced to 3/4 cup and glaze coats back of spoon, stirring often, about 25 minutes. (Can be made 2 days ahead. Cool glaze, then cover and chill. Rewarm over medium heat, whisking occasionally.)
Whisk in 3 tablespoons butter.
Season glaze with salt and pepper.
Prepare barbecue (medium-high heat).
Reserve 1/4 cup glaze in bowl.
Grill chicken until cooked through, basting with reserved glaze, about 5 minutes per side.