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1. Using stand mixer fitted with paddle, mix tapioca starch, 2 teaspoons salt, and baking powder on low speed until combined, about 30 seconds.
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2. Combine milk, oil, and butter in medium saucepan and bring to boil over high heat. With mixer running on low speed, working quickly, pour milk mixture over tapioca mixture and continue to mix on low speed until all ingredients are incorporated, about 3 minutes longer.
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3. Add 2 eggs and mix on low speed until dough comes together, turns shiny and sticky, and clings to sides of bowl, about 8 minutes, scraping down paddle and bowl halfway through mixing.
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4. Add Parmesan and Pecorino and mix on low speed until cheeses are incorporated, 30 to 60 seconds. Mix with rubber spatula to ensure mixture is fully incorporated. Remove bowl from stand mixer and press plastic wrap directly onto surface of dough. Refrigerate for at least 2 hours or overnight.
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5. Adjust oven rack to middle position and heat oven to 450 degrees F/230 degrees C. Stack 2 baking sheets and line top sheet with parchment paper. Divide dough into 8 balls (about 3 1/2 ounces each). To form rolls, lightly dampen your hands with water and roll balls between your palms until smooth. Evenly space rolls on prepared sheet.
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6. Whisk 1 egg, 1 teaspoon water, and 1/4 teaspoon salt together in small bowl. Brush egg mixture over tops of rolls. Place rolls in oven and immediately reduce oven temperature to 375 degrees F/190 degrees C. Bake for 20 minutes. Rotate sheet and continue to bake until rolls are deep golden brown and outer crusts are dry and crunchy, about 20 minutes longer. Transfer rolls to serving platter and let cool for 5 minutes. Serve.
