In a large bowl, mix together flour, sugar and salt. Using your hands, mix in half of the butter, squeezing the cubes between your fingers, until the mixture resembles a coarse meal. Add in the remaining butter, squeezing the cubes between your fingers once again, until the largest pieces of butter are approximately the size of a pea. Drizzle water over the dough and bring together, kneading 3 to 4 times. Flatten the dough into a disk, wrap tightly in plastic wrap and cool in the refrigerator for 30 minutes.
When ready to bake, preheat oven to 375 degrees. Line a rimmed baking sheet with parchment paper.
In a medium skillet over medium-high heat, add the olive oil. Once the oil is hot, add garlic and brown for approximately 1 minute. Add chicken pieces and cook until browned on all sides. Season with salt and pepper. Remove from heat and cool.
Roll out the galette dough into a 10” round. Spread the pesto sauce in an even layer on the dough, beginning approximately 1” in from the edges of the dough. Add the shredded mozzarella on top of the pesto in an even layer. Add the chicken pieces, followed by the halved cherry tomatoes.
Fold the pastry edge up and over the filling to create a 1” border. Brush the pastry border and edges with the melted butter.
Bake for 35 minutes. There will be some liquid from the cheese and tomato juices, which you’ll want to carefully drain off the top. Transfer the galette with the parchment paper onto a rack to cool for approximately 5 minutes before serving. Top with chopped fresh basil.