Step 1

Using cold water to make the dough keeps it from overheating in the food processor. Use block mozzarella, not fresh here.

Step 2

Process flour, sugar, yeast, and salt in food processor until combined, about 3 seconds. With processor running, slowly add cold water and oil and process until dough forms sticky ball that clears sides of bowl, 30 to 60 seconds.

Step 3

Transfer dough to counter and knead until smooth, about 1 minute. Shape dough into tight ball and place in greased bowl. Cover bowl with plastic wrap and let dough rise at room temperature until almost doubled in size, 2 to 2½ hours. (Alternatively, dough can rise in refrigerator until doubled in size, about 24 hours. Let come to room temperature, about 2 hours, before proceeding.)

Step 4

Turn out dough onto lightly floured 16 by 12-inch sheet of parchment paper and coat lightly with flour. Flatten into 8-inch disk using your hands. Using rolling pin, roll dough into 12-inch circle, dusting dough lightly with flour as needed.

Step 5

Roll bottom edge of dough 2½ inches in toward center. Rotate parchment 180 degrees and roll bottom edge of dough (directly opposite first rolled side) 2½ inches toward center. (Opposing edges of rolled sides should be 7 inches apart.)

Step 6

Roll ends of rolled sides toward centerline and pinch firmly together to form football shape about 12 inches long and about 7 inches across at its widest point. Transfer parchment with dough to rimmed baking sheet. Cover loosely with plastic and let rise until puffy, 30 minutes to 1 hour. Adjust oven rack to middle position and heat oven to 450 degrees.

Step 7

Combine mozzarella and feta in bowl. Fill dough with cheese mixture, lightly compacting and mounding in center (cheese will be piled higher than edge of dough). Bake until crust is well browned and cheese is bubbly and beginning to brown in spots, 15 to 17 minutes. Transfer sheet to wire rack. Add egg yolk and butter to cheese filling and stir with fork until fully incorporated and cheese is smooth and stretchy. Lift parchment off sheet and slide bread onto serving dish. Serve immediately.