Preparation Place quail eggs in small saucepan; add enough cold water to cover by 1 inch. Add 1 tablespoon vinegar. Bring to boil. Remove pan from heat; cover and let stand 5 minutes. Add ice cubes to pan; let stand 15 minutes to cool. Gently tap top and bottom ends of each egg on work surface to loosen shell. Starting at bottom (rounded) end of 1 egg, carefully remove shell, pinching and squeezing egg between fingers. Repeat with remaining eggs. Place peeled eggs in 1 1/2- to 2-cup glass jar with lid. Add remaining 1 cup vinegar. Close jar; refrigerate at least 1 day and up to 2 days. Add 1 tablespoon coarse salt. Close jar and shake gently; chill 24 hours. DO AHEAD: Can be made 1 week ahead. Keep chilled.
Preparation Place quail eggs in small saucepan; add enough cold water to cover by 1 inch. Add 1 tablespoon vinegar. Bring to boil. Remove pan from heat; cover and let stand 5 minutes. Add ice cubes to pan; let stand 15 minutes to cool. Gently tap top and bottom ends of each egg on work surface to loosen shell. Starting at bottom (rounded) end of 1 egg, carefully remove shell, pinching and squeezing egg between fingers. Repeat with remaining eggs. Place peeled eggs in 1 1/2- to 2-cup glass jar with lid. Add remaining 1 cup vinegar. Close jar; refrigerate at least 1 day and up to 2 days. Add 1 tablespoon coarse salt. Close jar and shake gently; chill 24 hours. DO AHEAD: Can be made 1 week ahead. Keep chilled.