Add strained squab stock and cook at a brisk simmer until reduced to sauce consistency, about 2 cups. This will take an hour or more. Season to taste with salt and pepper. Let sauce cool, cover surface with plastic wrap and refrigerate. Peel salsify, placing peeled stalks in water with 2 tablespoons lemon juice or white vinegar.
Slice salsify crosswise into thin rounds and dry on paper towels.
Melt 2 tablespoons butter in a large sauce pan. Add salsify and turn to coat with butter. Return oven to 450 degrees. Place remaining 2 tablespoons butter and 1 tablespoon oil in an oven-proof skillet large enough to hold the squab pieces in a single layer. Brown pieces on all sides, then remove breast pieces.