Please yeast in a measuring cup and add warm water to the 1/4-cup line. Pour mixture into a bread machine; add eggs. Add the remaining water, flour, oil, 1/3 cup sugar, salt, and vanilla extract. Run Dough cycle, 1 to 2 hours.
Mix 2/3 cup butter, brown sugar, and corn syrup together in a bowl. Spread caramel mixture into the bottom of a 9x13-inch baking pan.
Roll dough out onto a floured work surface. Spread 1/4 cup butter, 1/4 cup sugar, and cinnamon over the dough; add the caramel topping. Roll up the dough and cut into rolls using a thread. Place rolls on top of the caramel sauce in the pan. Cover with a kitchen towel; let rise until doubled in volume, about 1 hour.
Preheat oven to 350 degrees F (175 degrees C).
Bake rolls in the preheated oven until golden brown, about 30 minutes. Turn out immediately to avoid sticking.