Make the Balsamic Glaze - In a small saucepan, add the balsamic vinegar and honey. Stir to combine. Turn the heat to medium-high, bring to a boil, and reduce to a bubbling simmer over medium heat. Stir as needed and turn down the heat if it starts to boil rapidly.Reduce the mixture by half to about ½ to ⅓ cup. It should be slightly thickened, coating the back of a spoon. This will take about 7 to 10 minutes. Transfer to a small bowl and allow it to cool and thicken further. Refrigerate to speed up the process.
Season the Chicken - Flatten the chicken with a meat tenderizer and lightly season both sides with salt and pepper. Drizzle a small amount of olive oil on both sides to coat.
Grill the Chicken - Heat the grill to medium-high. Add a small amount of vegetable oil on a folded piece of paper towel, and grease the grates. Once hot, add the seasoned chicken breasts. Cover and cook for 4 to 5 minutes per side. The meat should reach 160 to 165°F (71 to 74ºC). Transfer to a clean place and cover to keep warm. Alternatively, cook the chicken on a grill pan.
Make the Bruschetta - In a medium bowl, combine the tomatoes, red onions, garlic, ¼ teaspoon salt, ⅛ teaspoon pepper, 1 teaspoon olive oil, oregano, and basil. Set aside.
Assemble the Sandwich - Toast the bread until golden brown and slightly crisp. Add ¼ cup arugula, 2 slices of cheese and chicken. Top with bruschetta and drizzle with balsamic glaze.
Notes
Large Chicken Pieces: If using two large pieces of chicken, slice in half to make four cutlets. Flatten to 1/2-inch thick pieces.
Storing: Store in an airtight container and refrigerate for up to one day.