Preheat oven to 350 degrees F (175 degrees C). Grease a baking sheet.
Combine yeast and 1/2 cup plus 2 tablespoons warm water in the bowl of a stand mixer fitted with the dough hook attachment. Add 2 tablespoons sugar, olive oil, and salt; mix until smooth. Add 2 cups flour and 1 teaspoon cinnamon; mix until just combined and dough forms a ball. Allow to rest, about 15 minutes.
Combine 1 cup water, 1 cup brown sugar, 1 teaspoon vanilla extract, and 1 teaspoon cinnamon in a saucepan; bring to a boil. Add apples; cook and stir until tender, 3 to 5 minutes.
Mix remaining 1/3 cup water and cornstarch together in a bowl until dissolved; pour into saucepan with apples. Stir continuously until filling is thickened, about 3 minutes; remove from heat.
Place dough on the baking sheet; stretch to cover most of the sheet. Poke the surface all over with a fork. Distribute 1/2 cup brown sugar over the dough; spread apple filling on top.
Beat butter in a bowl until fluffy. Stir in remaining 1/2 cup brown sugar, 2/3 cup flour, 1/2 cup white sugar, 1 teaspoon cinnamon, and 1/2 teaspoon nutmeg until topping is just combined. Sprinkle topping over pizza.
Bake in the preheated oven until golden, about 12 minutes.