Instructions

FOR THE DOUGH (see below for food processor method): In bowl of standing mixer, whisk flour, yeast, and salt to combine. Attach bowl and dough hook to mixer; with mixer running at medium-low speed, add olive oil, then gradually add water; continue to mix until mixture comes together and smooth, elastic dough forms, about 10 minutes. Lightly spray large bowl with nonstick cooking spray; form dough into ball, transfer it to bowl, cover bowl with plastic wrap lightly sprayed with nonstick cooking spray, and let rise in warm spot until doubled in size, 1 1/2 to 2 hours.

for the filling

While dough rises, stir together garlic, olive oil, and red pepper flakes in 8-inch skillet over medium heat until garlic is fragrant and sizzling and mixture registers 200 degrees on instant-read thermometer, 1 1/2 to 2 minutes. Transfer to small bowl and cool until warm, stirring occasionally, about 10 minutes.
In medium bowl, stir together cheeses, egg yolk, oregano, salt, black pepper, and cooled -garlic-pepper oil until combined; cover with plastic wrap and refrigerate until needed.
Adjust oven rack to lowest position, set pizza stone on oven rack, and heat oven to 500 degrees. Line baking sheet with parchment paper and spray parchment lightly with nonstick cooking spray. Turn risen dough out onto unfloured work surface. Divide dough in half, then cut each half into thirds. Gently reshape each piece of dough into ball. Transfer to baking sheet and cover with plastic wrap lightly sprayed with nonstick cooking spray. Let dough rest at least 15 minutes but no more than 30 minutes.
Cut eight 9-inch squares of parchment paper. Working with one piece of dough at a time and keeping other pieces covered, follow illustration 1 to roll dough into 9-inch round. Set round onto parchment square and cover with another parchment square; roll out another dough ball, set dough round on top of first, and cover with parchment square. Repeat to form stack of 3 rounds, covering top round with parchment square. Form second stack of 3 with remaining dough balls and parchment squares.
Remove top parchment square from first stack of dough rounds and place rounds with parchment beneath on work surface; if dough rounds have shrunk, gently and evenly roll out again to 9-inch rounds. Following illustrations 2 through 6, form calzones. With pastry brush, brush tops and sides of calzones with olive oil and lightly sprinkle with salt. Trim excess parchment paper; slide calzones on parchment onto pizza peel or rimless baking sheet, then slide calzones with parchment onto hot pizza stone, evenly spacing them apart. Bake until golden brown, about 11 minutes; use pizza peel or rimless baking sheet to remove calzones with parchment to wire rack. Remove parchment, cool 5 minutes, and serve. While first batch bakes, form second batch and bake after removing first batch.

food processor method

Made in an 11-cup cup food processor, the dough bakes up with slightly less chew than we like.
Alternative Step 1: Pulse flour, yeast, and salt to combine, about five 1-second pulses. While pulsing, add olive oil through feed tube, then gradually add water; continue pulsing until dough forms ball, then process until smooth and elastic, about 30 seconds. Turn dough out onto lightly floured work surface and knead by hand a few turns to form smooth, round ball. Transfer dough to oiled bowl and proceed from step 2.