Stir yeast, sugar, and water together in a bowl; let sit for about 10 minutes.

Stir in flour and mix until a dry, shaggy dough forms. Add salt, butter, and 2 tablespoons oil and mix to combine.

Knead with your hands in the bowl until dough comes together, then turn out onto the counter and continue to knead until a stiff but smooth dough forms, about 1 to 2 minutes.

Return dough to the bowl; rub the surface of the dough with a small amount of oil. Cover the bowl and let dough rise in a warm place until doubled in size, up to 6 hours. Alternately, place dough in the refrigerator to rise overnight.

Blend San Marzano tomatoes until smooth using a stick blender or in the jar of an electric blender. Measure out 1 1/2 cups of the tomato puree and place into a bowl. Stir in kosher salt, dried oregano, and cayenne pepper. Refrigerate until ready to make the pizzas.

Preheat the oven to 500 degrees F (260 degrees C). Drizzle 1 1/2 to 2 tablespoons oil into a 10- to 12-inch pizza pan.

Divide dough into 4 pieces and roll into balls. Keep dough covered when not using. Press 1 dough ball into a round disc and place on the prepared pizza pan. Press and stretch dough flat, almost all the way to the edge of the pan. Poke the dough all over with a fork.

Top the dough with about 1/3 cup sauce, spreading all the way to the edge. Top the entire surface with 2 ounces each of Cheddar cheese and Monterey Jack cheese.