Place the dates in a small bowl and cover with ½ cup boiling water. Soak until the dates are softened, about 5 minutes. Drain and set aside. Meanwhile, peel and chop the shallots finely. In a medium skillet over medium-high heat, heat the oil and saute the shallots, stirring often, until lightly browned, about 4 minutes. Add the thyme and cook another 1 minute. Chop the softened dates and add to the shallots. Remove from the heat and set aside.
Heat the oven to 350°F. Cut the chicken breasts in half through their thickness (that is, parallel to the surface of the cutting board). Place each resulting piece between two sheets of wax paper and pound until the chicken is ¼-inch thick.
In a small bowl, combine the goat cheese and Neufchâtel cheese with the salt and pepper. Spread the cheese mixture over the chicken cutlets. Top with the shallot mixture. Starting with a long edge, roll each cutlet into a tight cylinder and secure with a toothpick through the center. Place on a rimmed baking sheet, seam-side down, and bake the roulades until cooked through, about 18 to 22 minutes. Using tongs, transfer the roulades to a cutting board. When they have cooled slightly, remove the toothpicks and wrap each roulade with a slice of prosciutto. Cut crosswise into ½-inch slices and serve.