Make the dough in a stand mixer: Pour water, sugar, and yeast into the bottom of the mixer’s bowl and let stand for 10 minutes. Add oil to yeast mixture, then flours, then salt and use the machine’s dough hook to pull the mixture into a craggy ball. Knead on low for 5 minutes, scraping down as needed, until a smooth, elastic ball has formed. Briefly remove it from your mixer bowl, oil the bowl, and return the dough to it, covering the bowl with plastic wrap. Let rise for 1 1/2 to 2 hours; it should double. (Mine was done on the early end — for once.)
Meanwhile, and I mean right away so it has time to leisurely cool, make the sauce: Heat 2 tablespoons olive oil in the bottom of a medium-sized pot over medium, then add garlic, cook until it barely picks up color, and add pepper flakes and oregano, stir again. Add canned tomatoes (be careful — it’s going to splash up) and salt and stir to combine. Add basil, bring the mixture to a simmer, then reduce to a low simmer, for 10 minutes, stirring occasionally. Remove and discard basil. Adjust seasonings to taste. Set aside to cool to lukewarm or room temperature while dough rises.
Mix cheeses together in a large bowl and refrigerate until needed.
To make a stromboli/packet-like/scaccia shape: Heat oven to 400 degrees F. Line a large baking sheet with parchment paper. On a floured surface, roll your dough into the thinnest rectangle that you can, pulling and stretching it as needed. You’re looking for 1/16-inch thickness; the longer sides should be parallel to you.
Bake the stromboli/packet/scaccia shape: For about 1 hour, until deeply browned all over and charred in some spots. Rotate the pan as needed for even coloring. Let stand at room temperature for 10 minutes before cutting into squares with a serrated knife, and serving.
To make a pizza babka: Heat oven to 400 degrees F. Line a loaf pan with a sling of parchment paper. On a floured surface, roll your dough into the thinnest rectangle that you can, pulling and stretching it as needed. You’re looking for 1/16-inch thickness; the longer sides should be parallel to you.
Bake the pizza babka: For about 1 hour, until deeply browned all over and charred in some spots. Rotate the pan as needed for even coloring. Immediately invert the pie onto a rack, remove the loaf pan and parchment paper, and let the pie cool in this position for 10 minutes. Invert the pie right-side-up before slicing with a serrated knife to serve.
Both shapes: Reheat fantastically. I wrap them in foil in a 350 degree oven until warmed through, about 15 minutes. I’d expect them to freeze well too.