In a medium bowl, stir the shredded chicken with the pesto to coat. 

Lay a flour tortilla and add half of the chicken pesto mixture to one half of the tortilla, leaving a ½ border on the round side of the tortilla. 

If you want extra cheese, lay two slices of provolone on the tortilla first, then the pesto chicken mixture. 

Top the pesto chicken two slices of tomato, and then two more slices of provolone cheese. 

Fold the naked side of the tortilla over the filling and lightly press. 

In a 12-inch nonstick skillet, heat 1/4 cup of grapeseed or vegetable oil to the skillet. Heat the oil over medium heat until it’s shimmering. 

While the oil is heating, prepare the second quesadilla. 

Then, carefully lay one folded quesadilla in the skillet. Lightly press the quesadilla to ensure the tortilla surface has full contact with the pan. Lightly tilt the pan from side to side and top to bottom to swirl the oil around the quesadilla. 

Let the first side of the quesadilla cook for about 2 minutes, until it’s golden.

When it’s time to flip, use two nonstick-safe spatulas (one under the quesadilla and one on top) to gently flip the quesadilla over. Lightly press the quesadilla into the pan and swirl the oil again. Let the other side cook for about 2 minutes, until it’s golden brown.

When golden on the second side, cut the heat and transfer the quesadilla to a wire rack to cool slightly.

Cook the second quesadilla following the same method. 

Transfer the quesadillas to a cutting board and slice into three wedges.