Combine 1 cup flour, undissolved yeast, sugar and salt in a large bowl. Add very warm water and oil; mix until well blended, about 1 minute. Gradually add enough remaining flour to make a soft dough. Dough should form a ball and will be slightly sticky. Knead** on a floured surface, adding additional flour if necessary, until smooth and elastic, about 4 minutes.
Pat dough with floured hands to fill greased pizza pan (with a rim) or baking sheet. OR roll dough on a floured counter to 12-inch circle; place in greased pizza pan or baking sheet. Form a rim by pinching the edge of the dough. Use a fork to dock the crust every inch or so.
Arrange apples in concentric circles on crust. Combine brown sugar, corn starch and cinnamon in a small bowl; mix well. Stir in butter, corn syrup and vanilla. Dollop over apples. Sprinkle with peanuts.
If using a pizza pan, place a larger baking sheet under the pan to catch any boil overs. Bake on center oven rack for 16 to 21 minutes, until crust is lightly browned and apples are tender. Let pizza rest 10 minutes before serving.
*If you don’t have a thermometer, water should feel very warm to the touch.
**To knead the dough, add just enough flour to the dough and your hands to keep the dough from sticking. Flatten dough and fold it toward you. Using the heels of your hands, push the dough away with a rolling motion. Rotate dough a quarter turn and repeat the “fold, push and turn” steps. Keep kneading dough until it is smooth and elastic. Use a little more flour if dough becomes too sticky, always working the flour into the ball of dough.