Toss quail with black pepper and half of the salt. Cover and chill at least 3 hours.
Whisk eggs, buttermilk, and hot sauce in a large bowl. Whisk flour and remaining salt in another large bowl.
Fit a Dutch oven with thermometer; pour in oil to measure 2". Heat over medium-high heat until thermometer registers 325°. Pat quail dry. Working with 1 piece at a time, dredge in flour mixture, shaking off excess, then dip in buttermilk mixture, letting excess drip back into bowl. Dredge again in flour mixture and place on a baking sheet.
Fry quail, turning occasionally, until skin is deep golden brown and crisp. Transfer to a clean wire rack set inside a baking sheet. Let oil cool slightly.
Whisk cayenne, brown sugar, chili powder, garlic powder, and paprika in a medium bowl; carefully whisk in 1/4 cup frying oil. Brush fried quail with spicy oil. Serve with bread and pickles.