Prepare ingredients and equipment: measure all ingredients. If using active dry yeast, heat 1/2 cup of the warm water to 110–115°F. If using instant yeast, heat 1 1/2 cups warm water to 120–130°F. Fit an electric stand mixer with the paddle attachment.
For active dry yeast: in the mixer bowl whisk together 2 tsp active dry yeast, 2 tsp granulated sugar and the 1/2 cup warm water (110–115°F). Let rest 5–10 minutes until the mixture becomes foamy and bubbly.
For instant yeast: do not proof. Add 2 tsp instant yeast and 2 tsp granulated sugar directly to the mixer bowl with 3 cups of the measured flour (see step 5) before adding the warm water.
Add to the mixer bowl (with either yeast method): the remaining warm water (for active dry yeast add the remaining 1 cup; for instant yeast add all 1 1/2 cups total as noted in step 1), 3 Tbsp olive oil, and 1 1/4 tsp salt.
Add 3 cups of the 3 1/2 cups (17 oz) unbleached all-purpose flour or bread flour to the mixer bowl. Mix on low speed with the paddle attachment just until the dough comes together and no large dry patches remain.
Switch to the dough hook. Add the remaining 1/2 cup flour to the dough and begin kneading on low speed. If the dough is very sticky, add more flour 1–2 Tbsp at a time as needed (use sparingly) until the dough is not overly tacky.
Continue kneading on low speed about 4 minutes, until the dough is soft, supple, and pulls away from the sides of the bowl (it may still stick slightly to the bottom). The dough should be slightly sticky but manageable.
Lightly coat a large bowl with olive oil (or use olive oil cooking spray). Transfer the dough to the oiled bowl and turn it once to coat the top with oil.
Cover the bowl tightly with plastic wrap and let the dough rise in a warm place until doubled in volume, about 1 1/2 hours. Alternatively, refrigerate the covered dough for up to 24 hours for a slower rise (if refrigerating, allow the dough to come to room temperature for about 1 hour before shaping).
Near the end of the rise, place a pizza stone in the oven and preheat the oven to 475°F. Allow the stone to preheat for 30–45 minutes.
After the dough has doubled, press it down to deflate slightly and divide it into two equal portions. If you plan to save one portion for later, freeze that second portion now (following your preferred freezing method).
If using both portions right away, place each portion into its own lightly oiled bowl, cover, and let rest 15 minutes to relax the gluten.
On a lightly floured surface, stretch and shape each portion into a round about 11–14 inches in diameter, depending on how thick or thin you prefer the crust.
Transfer each shaped dough to a sheet of parchment paper just large enough to fit the pizza. Brush the top of each crust with olive oil (about 1 Tbsp each).
If you prefer a fluffier crust, let the shaped and oiled dough rest an additional 10 minutes before adding toppings.
Top the pizza: spread sauce (about 2/3 cup), sprinkle with cheese (about 2 cups mozzarella), and add desired toppings.
Slide the parchment with the topped pizza onto the preheated pizza stone and bake until the crust is golden brown and the pizza is cooked through, about 11–14 minutes.
Remove the pizza from the oven, slice, and serve warm. Repeat the shaping and baking process with the second dough portion if using.