Tip:If you are unsure if the food is done, open one of the packets to check. If it’s not ready, reseal it as best you can and return all the packets to the oven.
Position racks in the upper and lower thirds of the oven and heat the oven to 375°F. Meanwhile, prepare foil packets. Start with four 12x20-inch pieces of regular or heavy-duty foil. Fold them in half to make 12x10-inch rectangles.
Thoroughly pat the chicken breasts dry and season with salt and pepper. Heat 1 Tbs. of the olive oil and 1 Tbs. of the butter in a 12-inch skillet over medium-high heat. When the butter stops foaming, add the chicken and cook until browned on both sides, 3 to 4 minutes per side. Transfer to a plate. Return the pan to the heat, add the vermouth, scrape up any brown bits from the bottom of the pan with a wooden spoon, and cook until reduced to about 2 Tbs. Immediately remove from the heat and set aside.
Arrange the foil sheets on a work surface. Coat the inside of each with the remaining 1-1/2 Tbs. butter, leaving a 3-inch border. In a medium bowl, toss the celery and prosciutto with 1 tsp. of the olive oil, 1/2 tsp. salt, and 1/4 tsp. pepper, breaking up any prosciutto clumps with your fingers. Distribute the celery mixture among the sheets, arranging it on one side in a layer about the size of a chicken breast. Scatter the capers over the celery. Put the chicken breasts on top and drizzle each with the reserved the vermouth. Scatter the tomatoes over the chicken. Top each with a bay leaf and drizzle with the remaining 2 tsp. olive oil. Fold the other of the foil over the ingredients, then tightly fold or crimp the edges to seal.
Transfer the packets to two large, rimmed baking sheets and bake until the packets are puffed and fragrant, 19 minutes. Serve immediately.
Creamy polenta or mashed potatoes would be a perfect accompaniment.