Pour olive oil into large skillet. Add squash; cook and stir over MEDIUM-HIGH heat about 5 minutes or until golden brown. Add onion and thyme to skillet; season to taste with salt and pepper. Cook and stir 2 minutes or until onions are translucent. Remove from heat; stir in cranberries. Cool.
Unroll pizza dough onto a lightly floured baking sheet. Roll out to into a large rectangle (about 13x10-inches). Sprinkle half the sausage down center of each dough. Top each with half of the squash mixture.
Use a sharp knife to make 7 diagonal cuts in exposed pastry on each side of filling, stopping ½ inch from filling. Fold strips over filling, alternating sides until filling is enclosed. Repeat with remaining dough and ingredients.
Beat egg with 2 teaspoons water. Brush dough with egg. Sprinkle with caraway seeds.
Bake 20-minutes or until braids are golden brown. Cut each bread into 6 slices to serve.