Pour the water in a small bowl and heat in the microwave about 11 seconds. You should be able to dip your finger in the bowl and not feel it's either too hot or too cold. Once the temperature is just right, add the yeast and sugar and stir. Set aside to proof the yeast, about 5 minutes. Once it's proven itself, add the vegetable oil. Set aside.
In a large bowl, combine the flours, salt, and ground flax seed. Stir together, until well combined.
Pour the yeast mixture into the bowl with the flour and stir vigorously for several minutes. This is a nice, pliable dough so I was able to knead it right in the bowl for 5 - 7 minutes. You want to reach the point where you can make an impression with your thumb in the middle of the dough and the dough raises back. Pour the olive oil over the dough and turn it several times to ensure both the dough and the bowl are coated with oil. Cover and set aside to rise. It will need about 2 hours to rise.
Once the dough is ready, cover it with a large sheet of waxed paper and use a rolling pin to spread it out to the dimensions of your pan. This will fit a 12" deep dish pizza pan. Make sure you allow the dough to fit up the sides of the pan a bit to create a sort of deep-dish pizza bowl! Set aside to prepare your toppings.
Whenever you're ready to prepare your topping, begin by heating your oven to 450F.
Next, place the thinly sliced vegan Jack cheese across the bottom of the dough. Cover that with the shredded vegan mozzarella. Spread it out so you don't see any of the pizza dough coming through.
Place the chopped tomatoes in a food processor along with the basil, oregano, and vegan parmesan. Pulse for just a few seconds. You want the tomatoes to be fairly ground, but still with lots of texture. Spread the tomato mixture over the cheese.
Spread the chopped slices of vegan pepperoni evenly over the top of the tomato mixture.
Sprinkle with additional vegan parmesan.
Place in your prepared oven for 20 - 25 minutes. The dough should be showing through with a nice golden color. Allow the pizza to cool for about 10 minutes before slicing and serving.