Heat faucet water to 110 degrees. If you don't have a cooking thermometer, water should feel warm to the touch. Measure 1¼ cup.
Add ½ tsp sugar and 1 tsp yeast to the warm water. Gently stir to combine. Let sit for 10 minutes until bubbles form on top.
Combine the 2 cups AP flour, 2 cups bread flour, and 1 tsp salt in a large bowl.
Make a well in the middle of the mixture and add 4 tbsp extra-virgin olive oil. Stir gently with a wooden spoon to combine.
Add the bubbled yeast mixture and mix until the flour is mostly incorporated and a shaggy dough begins to form.
Empty the dough onto a clean work surface (I use a large wooden cutting board) and knead for 10 minutes. The dough should look smooth and feel soft and bouncy. If you press your finger into the dough, it should bounce back.
Lightly coat a clean bowl with olive oil and place the dough inside, gently rolling the dough in the bowl to cover it with oil. Cover with plastic wrap. Place in a warm part of your kitchen and let it rise at least 2 hours (5 hours is a good target). Dough can be left to rise for a total of 24 hours.
Punch down the dough, move it to a lightly floured work surface, and knead for 1-2 minutes. Cut into 2 equal portions and shape into rounds.
Lightly dust the dough with flour, cover with a kitchen towel, and set aside to rise for 20-30 minutes. Gather and prepare any toppings during this time.