Preheat your oven to 400°F (200°C). Place the 2 flatbread pizza bases on a baking sheet and brush each one with 1 tablespoon of olive oil to coat the surface evenly.
Pour 1/2 cup of balsamic vinegar into a small saucepan and set it over medium heat. Bring it to a boil, then reduce the heat to low and let it simmer until the vinegar has reduced by about 1/3 and coats the back of a spoon, forming a thick glaze. Remove from heat and set aside to cool and thicken. I like to make extra balsamic glaze and keep it in the fridge for salads throughout the week!
Heat the remaining 1 tablespoon of olive oil in a skillet over medium-low heat. Add the diced or sliced shallots and sauté for 4-5 minutes until they are lightly browned. Sprinkle with a pinch of sea salt. Remove the skillet from the heat, drizzle in the 2 tablespoons of balsamic vinegar, and toss to deglaze the pan, letting the shallots absorb the flavor.
Arrange the sliced burrata or mozzarella cheese evenly over each prepared flatbread. Top with the caramelized and balsamic-glazed shallots (from Step 3), followed by the chopped or sliced cooked chicken breast, pecan halves, halved fresh strawberries, and another light sprinkle of sea salt.
Bake the assembled flatbreads in the preheated oven for 10-12 minutes, or until the cheese has melted and the edges are golden. Remove from the oven and drizzle with the balsamic reduction (from Step 2). Top with fresh arugula and basil leaves as desired. For a more vibrant finish, I like to add a generous handful of arugula just before serving.