Prepare the balsamic glaze: Heat 1/4 cup balsamic vinegar in a small saucepan over medium heat. Simmer until syrupy and reduced to about a tablespoon, 8 to 10 minutes.
Grill the chicken: Preheat an outdoor grill or indoor grill pan to medium high (about 425 degrees F). Slice the chicken breast in half length-wise so that you have 2 chicken cutlets. Brush with the remaining 2 tablespoons balsamic vinegar and 1 tablespoon olive oil. Grill on each side until the chicken is cooked through and reaches an internal temperature of 165 degrees F, about 6 to 8 minutes per side. Remove to a plate, cover, and let rest 5 minutes. Cut into thin slices, then re-cover and set aside.
Prepare the rest of your toppings: Peel, pit, and slice the avocado (cover it with plastic or a squirt of lemon juice to prevent browning); cut the strawberries into 1/4-inch-thick rounds or slices; crumble the goat cheese; and slice the basil into thin ribbons.
Grill the naan and assemble the pizzas: If using a grill pan indoors on the stove, heat the grill pan over medium high. Brush both sides of one of the naan with olive oil and place it in the grill pan, pressing down on the top of it with a spatula to sear. Cook on the first side until just beginning to char, about 2 to 3 minutes. Flip and cook on the second side, just until you have light browning, 30 seconds to 1 minute, then remove to a sheet pan or large plate. Repeat with second naan. If using an outdoor grill, ensure your grill plates are very clean (I recommend heating the grill to very high heat, then brushing the grates to remove any residue). Reduce the heat to medium, about 350 degrees F. Brush both sides of the naan with oil, place them on the grill, and close the lid. Let cook 2 to 4 minutes, until the first side has light grill marks. Flip, close the lid, and cook 30 seconds to 1 additional minute, until the second just side begins to brown. Remove to a sheet pan or large plate.
Arrange the strawberries, grilled chicken, goat cheese, and avocado slices on the charred side of the naan. Carefully place it back on the heat (if you are using a grill pan, reduce the heat to medium). Cover and let cook until the cheese is warmed through and the bottom finishes browning, about 2 additional minutes. Remove from heat. Drizzle with the balsamic syrup and sprinkle with fresh basil. Serve immediately.
This recipe is best enjoyed the day it is made. For quicker, easier prep, you can cook the chicken and slice the strawberries up to 1 day in advance and store separately in the refrigerator. The balsamic reduction will keep in the refrigerator for about 1 week.
*If you don't care for goat cheese, you can substitute crumbled feta or mozzarella. I suspect gorgonzola would also be delicious.