Whisk together tomato sauce, tomato paste, basil and 1T sugar and oregano in a small bowl; set aside.
Toss cheeses with liquid smoke in a medium bowl; set aside.
Combine flour, cornstarch, 2 tsp sugar, baking powder and salt in a large bowl. Combine water and olive oil in measuring cup. Stir water mixture into flour mixture until dough starts to come together. Turn dough onto slightly floured surface and knead 3 or 4 times, until cohesive.
Adjust oven rack to lower-middle position, place pizza stone (or inverted baking sheet) on rack and heat oven to 475 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" circle, rotating parchment as needed. Lift parchement and pizza off work surface onto inverted baking sheet (not one in the oven).
Top each piece of dough with half of the sauce and half of cheese. Carefully pull parchment off of baking sheet onto hot oven stone. Bake until underside is golden brown and cheese is completely melted, 9-12 minutes. Remove pizza and parchment from oven. Transfer pizza to cooling rack and let cool a few minutes. Assemble and bake second pizza. Cut into 2" squares. Serve.
Note - dough can be made in advance. Once mixed tightly wrap dough in plastic and refrigerate up to 2 days.
Variation - If you would like to add pepperoni arrange them on a paper towel lined plate, cover with paper towl and microwave 2 minutes. Let cool. Add to pizza either before or after cheese.