Instructions

for the dough

Whisk flour and yeast together in bowl of stand mixer. Add milk and eggs. Using dough hook on medium-low speed, mix until cohesive dough forms and no dry flour remains, 2 to 4 minutes, scraping down sides of bowl with rubber spatula as needed. Turn off mixer, cover bowl with dish towel or plastic wrap, and let dough stand for 15 minutes.
Add sugar and salt to dough and knead on medium speed until incorporated, about 30 seconds. Increase speed to medium-high and, with mixer running, add butter, 1 piece at a time, allowing each piece to incorporate before adding next, about 3 minutes total, scraping down dough hook and sides of bowl as needed. Continue to knead on medium-high speed until dough begins to pull away from sides of bowl, 7 to 10 minutes, scraping down bowl as needed.
Transfer dough to greased large bowl. Cover tightly with plastic and let rise until slightly puffy, about 1 hour. Refrigerate until firm, at least 2 hours or up to 24 hours.

for the filling

Combine oil, garlic, oregano, and pepper flakes in bowl and microwave until fragrant, 45 to 60 seconds, stirring halfway through microwaving; let cool completely. Stir in tomato paste and vinegar and season with salt and pepper to taste.
Grease 8½ by 4½-inch loaf pan. Press down on dough to deflate, then transfer to lightly floured counter and press and roll dough into 18 by 7-inch rectangle, with short side parallel to counter edge. Spread filling over dough, leaving ½-inch border, then sprinkle basil, provolone, and ¼ cup Pecorino over filling. Roll dough away from you into firm cylinder and pinch seam and ends closed. Wrap in plastic; transfer to rimmed baking sheet; and refrigerate until firm but still supple, about 30 minutes.
Transfer chilled dough log to lightly floured counter with short side facing you. Using bench scraper or sharp knife, cut log in half lengthwise. Turn dough halves cut side up and arrange side by side. Gently stretch each half into 14-inch length, then pinch top 2 ends of strips together to seal. Twist strips tightly by gently lifting left strip of dough over right strip of dough. Repeat, keeping cut side up, until all of dough is twisted. Pinch bottom ends together to seal.
Transfer loaf, cut side up, to prepared pan, reshaping as needed to fit into pan. Cover loaf loosely with greased plastic and let rise until almost doubled in size, 1 to 1½ hours (top of loaf should rise about 1 inch over lip of pan).
Adjust oven rack to middle position and heat oven to 350 degrees. Gently brush loaf with egg wash; sprinkle with remaining ½ cup Pecorino; and bake until loaf is deep golden brown and registers 190 to 195 degrees, 40 to 55 minutes, rotating pan halfway through baking. Let loaf cool in pan on wire rack for 1 hour. Remove loaf from pan and let cool completely on wire rack, about 3 hours, before serving.