Place a wide saute pan over medium-high heat and add the balsamic vinegar. Bring to a boil and allow the vinegar to reduce, stirring occasionally. Allow the vinegar to reduce until it is thick and syrupy enough to coat the back of a spoon, about 8-10 minutes. Remove from the heat, pour into a cool bowl and set aside for later. Clean the pan and return it to the stovetop.
Place the chicken thighs between cling film. Using a kitchen mallet, pound thighs so that they are each about 1″ thick. Sprinkle evenly with the salt.
Add the olive oil to the pan and heat on medium-high until shimmering. Add the chicken thighs in one layer and fry until golden brown on the bottom, then turn and brown the other side.
Layer the chicken with 1-2 leaves of basil, then the tomatoes, then the slices of fresh mozzarella.
Turn off the heat of the stovetop and place the pan in the hot oven.
Bake for 10-12 minutes, or until the cheese has fully melted and is beginning to brown.
Remove the pan from the oven and drizzle with the reserved balsamic vinegar glaze
Serve hot.