In a medium bowl, combine flour, egg yolk, sour cream and water. Work into a soft elastic dough. Knead 15-20 minutes. Cover, set aside.
In a medium bowl, combine flour, butter into a smooth soft dough. Divide dough into 3 balls. Cover dough balls, set aside.
On a lightly floured board roll out DOUGH 1 into a rectangle about 1/4 inch thick. Place 1 ball of DOUGH 2 on center of rolled dough rectangle.
Gently press dough ball with your hands or rolling pin until dough ball is 1/4 inch thick and covers a circular portion of DOUGH 1. Fold up both short sides of DOUGH 1 over rolled DOUGH 2 so ends meet in center. Re-roll resulting dough combination into a rectangle 1/4 inch thick. Repeat with remaining 2 balls of DOUGH 2. Roll dough jelly-roll style. Wrap dough in foil; refrigerate overnight.
Halve dough crosswise. On a lightly floured board, roll out 1/2 of dough into a rectangle 1/4 to 3/8 inch thick. Using a round 2 inch cutter, cut dough rounds. Arrange 16 rounds on greased baking sheet, allowing room for spreading. Press leftover dough scraps into a ball. Roll to same thickness; cut as many dough rounds as possible. Repeat r process with remaining dough until a total of 48 rounds are cut.
With a 1 inch round cutter, cut out centers of 32 dough rounds not on baking sheets, forming doughnut shapes. Using a pastry brush, apply beaten egg white to top surface of 1 uncut dough round on baking sheet. Carefully postion 1 doughnut cutout directly on top of brushed dough round. Lightly press doughnut cutout down with your fingertips. Brush egg white on top of doughnut cutout. Place a second doughnut cutout on top of the first. Pat lightly with your fingers.
Repeat process until all dough rounds on baking sheet have 2 doughnut cutouts glued on. Brush beaten egg yolk on top of doughnut cutouts. Bake 15 minutes or until golden; let cool. Melt butter in skillet, add mushrooms, saute until nearly tender. In bowl, combine half and half, flour, salt, pepper; stir mixture into sauteed mushrooms. Simmer 5 minutes. Set aside to cool. FIll each doughnut round with about 1 teastpoon of cooled mushroom mixture. Bake 8-10 minutes. Serve hot.
THIS IS A LOT HARDER TO DESCRIBE THAN TO DO!