Instructions

Whisk flour, thyme, sage, fennel, yeast, and salt together in bowl of stand mixer. Whisk milk, sugar, and egg yolks in 4-cup liquid measuring cup until sugar has dissolved. Fit mixer with dough hook and slowly add milk mixture to flour mixture, mixing on low speed until cohesive dough starts to form and no dry flour remains, about 2 minutes, scraping down dough hook and sides of bowl as needed. Increase speed to medium-low and add softened butter, 1 piece at a time. Knead for 4 minutes, scraping down dough hook and sides of bowl occasionally. Continue to knead until dough is smooth and elastic and clears sides of bowl, 10 to 12 minutes.
Transfer dough to lightly floured counter and knead by hand to form smooth, round ball, about 30 seconds. Transfer dough seam side down to lightly greased large bowl, cover with plastic wrap, and let rise until increased in size by about half, 1½ to 2 hours. Place in refrigerator until dough is firm, at least 1 hour or up to 24 hours. (If dough is chilled longer than 1 hour, let rest at room temperature for 15 minutes before rolling out.)
Press down on dough to deflate. Transfer to clean counter, cut into 4 pieces, then knead each piece to form smooth ball. Cover loosely with plastic wrap. Transfer 1 dough ball to double layer of parchment paper (keep remaining dough balls covered) on counter and press and roll dough into 12-inch circle on parchment. Shingle half of prosciutto evenly over dough. Using parchment paper as sling, transfer dough and parchment to cutting board.
Transfer second dough ball to lightly floured counter and press and roll into 12-inch circle. Carefully top prosciutto-topped dough round with second dough round. Microwave fig jam until loosened, about 30 seconds, then evenly brush over top dough round.
Transfer third dough ball to lightly floured counter and press and roll into 12-inch circle. Carefully top fig-topped dough round with third dough round. Shingle remaining prosciutto over top. Transfer final dough ball to lightly floured counter and press and roll into 12-inch circle, then lay dough round on top of stack.
Place 3-inch wide cookie cutter or overturned bowl in center of dough (do not press into dough). Using sharp knife positioned at 12 o’clock, cut through all 4 layers of dough and filling, starting from edge of cookie cutter and finishing at edge of dough round (be sure to cut firmly through prosciutto layers). Repeat cut on opposite side of cookie cutter, at 6 o’clock. Continue to repeat cuts around dough round to make 16 evenly spaced cuts around cookie cutter. Remove cookie cutter.
Grasping ends of 2 adjacent strips, twist strips twice, then twist another half turn and pinch ends together to seal. Repeat with remaining strips. Using parchment as sling, carefully lift star and place on rimmed baking sheet (use caution as some cuts may have sliced through top parchment layer). Cover loosely with greased plastic and let rise until puffy, 40 to 60 minutes.
Adjust oven rack to middle position and preheat oven to 350 degrees. Brush pinwheel with egg wash and bake until golden brown, 30 to 40 minutes (if edges brown faster than center, shield edges with aluminum foil). Transfer sheet to wire rack and let cool for 15 minutes. Serve warm or at room temperature.