Place the carrot juice in a medium microwave-safe bowl. Microwave on high power for 1 minute or until nice and warm (not hot).
Combine the flour, yeast and salt in a large bowl. Stir to combine.
Make a well in the center of the flour mixture and add the warm carrot juice and the honey. Stir well, until the dough is shaggy and there is no more dry flour. Add the olive oil and stir until combined. The surface will be a slick with oil.
Cover the boil tight with plastic wrap or a large plate. Set aside for 30 minutes.
After 30 minutes, remove the cover and, using either a large spatula or slightly we hands, pick up one side of the dough, stretch it, and then fold it over the dough, tucking it down on the other side. Turn the bowl one-quarter turn and repeat the stretch and fold. Repeat the stretch and fold 3-4 more times, turning the bowl one-quarter each time. (Don’t skip this step! This helps aerate the dough which creates those beautiful artisan air pockets.)
Cover the bowl and let the dough rise for one hour.
During the one-hour rise, use a piece of paper toweling and rub 1 teaspoon of butter over the interior of 2 9-inch round baking pans. Line the bottom of each pan with a round of parchment paper. Rub ½ teaspoon of butter over the parchment paper in each pan. Drizzle each pan with 1 tablespoon extra virgin olive oil. Set aside.
After the one-hour rise, divide the dough into the two prepared pans, flipping once to coat the dough with the oil. Use the same stretching and folding described above to form each ball of dough into a rough circle. Flip the dough so the folded sides are down. Flatten the dough with your hands in the pans. It won’t cover the entire pan at this point but flatten it out as much as you can. Then cover the pans with the plastic wrap or 2 plates. Set aside to rise again for one more hour.