Sauce and cheese:
Whisk tomato sauce, tomato paste, basil, sugar, and oregano together in small bowl; set aside. Toss cheeses with liquid smoke in medium bowl; set aside.
Combine flour, cornstarch, sugar, baking powder, and salt in large bowl. Combine water and olive oil in liquid measuring cup. Stir water mixture into flour mixture until dough starts to come together. Turn dough onto lightly floured surface and knead 3 or 4 times, until cohesive.
Adjust oven rack to lower-middle position, place baking stone (or inverted baking sheet) on rack, and heat oven to 475 F degrees.
Divide dough into 2 equal pieces. Working with 1 piece of dough at a time, press into small circle and transfer to parchment paper dusted lightly with flour. Using rolling pin, roll and stretch dough to form 12-inch circle, rotating parchment as needed. Lift parchment and pizza off work surface onto inverted baking sheet.
Top each piece of dough with half of sauce and half of cheese. Carefully pull parchment paper and pizza off baking sheet onto hot baking stone. Bake until underside is golden brown and cheese is completely melted, 9 to 12 minutes.
Remove pizza and parchment from oven. Transfer pizza to wire rack and let cool briefly. Assemble and bake second pizza. Cut into 2-inch squares. Serve.