Separate each quail into halves by cutting each bird down the middle of the breasts and back. Poke a toothpick through the thigh and breast to make them easier to handle when frying. Place the quail in a large bowl and submerge them in buttermilk. Cover and refrigerate overnight.
To make the tamarind sauce, sauté garlic and 1 tsp. oil in a medium saucepan over medium-low heat for 30 seconds until fragrant. Then add the remaining sauce ingredients and stir to dissolve until there are no lumps. Next, push the sauce through a fine-mesh strainer to remove remaining tamarind seeds and shells. Season to taste, and set sauce aside.
Fill a heavy-bottomed, 10-inch frying pan halfway with oil and heat to 325°. Meanwhile, combine 1 cup flour, ¼ tsp. paprika, ¼ tsp. ground black pepper, and 1 tsp. fine sea salt in a wide, shallow bowl. Remove quail from the buttermilk and set on a large rimmed cookie sheet; do not pat dry. Generously season quail with salt.
When the oil approaches 325 degrees, generously coat the wet birds in the flour mixture. Carefully lay them breast-side down in the hot oil without crowding the pan. Fry your quail until they’re golden brown and then flip them to fry the other side. Repeat with the remaining quail, adjusting heat as needed. Drain the cooked quail on a frying rack and keep them warm in the oven.
If needed, add more water to thin out your tamarind sauce. Before serving, toss or brush the fried quail with the sauce. Warn your dining guests about the toothpicks.