To prepare dough: In bowl of a food processor fitted with a metal blade, add flour, butter, shortening, powdered sugar, yeast and salt and pulse until mixture resembles coarse meal. Transfer mixture to a large bowl and stir in sour cream until a dough forms (if dough appears shaggy and dry after adding sour cream, add up to 2 tablespoons ice water until dough is smooth). On a lightly floured work surface, divide dough in half. Pat each half into a 7- by 3-inch rectangle and wrap in plastic wrap. Refrigerate dough for 30 minutes, then freeze until firm, about 15 minutes.
Line two baking sheets with parchment paper. On a lightly floured work surface, using a lightly floured rolling pin, roll one dough half into a 28- by 5-inch rectangle (dough will be about 1/4 -inch thick). Cover bottom (long) half of dough with half of filling, leaving a 1/2-inch border around bottom and side edges. Brush edge of uncovered dough with water, fold dough over filling and pinch seams closed. Shape into an oval, tuck one end inside the other and pinch to seal. Transfer to prepared baking sheet, cover with plastic wrap and refrigerate at least 4 hours or up to 12 hours. Repeat with remaining dough and filling.
When ready to bake, adjust oven racks to upper-middle and lower-middle positions and preheat oven to 350 degrees. Discard plastic wrap, brush kringles with beaten egg and bake until golden brown, 40 to 50 minutes, switching and rotating sheets halfway through baking. Remove from oven, and slide parchment paper and kringles to a wire rack and let cool about 30 minutes.
To prepare glaze: In a medium bowl, whisk powdered sugar, milk and vanilla extract until smooth. Drizzle glaze over kringles and let glaze set, about 10 minutes. Serve warm or at room temperature.