Sprinkle over the paprika over both sides of the chicken and season with salt and pepper. Heat 1 tablespoon oil in an ovenproof frying pan over medium-high heat. Cook the chicken in batches, turning, for 3-4 minutes or until browned. Transfer to a plate.
Reduce the heat to medium. Add remaining oil to pan with the onions and garlic. Cook, stirring, for 2 minutes or until soft. Add the vinegar and sugar. Cook, stirring, for 2 minutes. Add the cherry tomatoes, bring to the simmer. Return the chicken.
Press a piece of baking paper onto the surface and cover with a lid. Cook on medium-low for 10 minutes. Add the olives and broccolini, press the baking paper onto the surface to cover, and cook further 3-4 minutes until broccolini is bright green. Remove the cover.
Preheat a grill on medium-high. Scatter the bocconcini over the chicken. Grill for 3-4 minutes until the cheese has softened. Scatter over the basil. Serve with garlic bread.
Tip Tip Caprese chicken is also delicious served over pasta, or mashed potato.