Preheat oven to 180 degrees C. Spread 1 tablespoon of olive oil on a baking tray then set aside.
Heat 1 tablespoon of olive oil in a frying pan over medium heat. Stir in the eschalots and cook until they turn translucent, about 3 minutes. Stir in the garlic, thyme, salt and pepper; cook and stir an additional 2 minutes. Transfer the eschalot mixture to a bowl. Mix in the goats cheese, dates and basil; stir until well combined.
With a sharp knife, cut a 3cm long slit into the thick side of each chicken breast. Work your fingers into the slit and expand the slit to form a pocket in the breast meat. With your fingers or a spoon, stuff each chicken breast with about 1/4 cup of the goats cheese mixture. Wipe off any cheese mixture from the outside of the chicken breast and wrap each breast in a slice of prosciutto so that the pocket opening is covered. Place the chicken breasts, seam sides down, onto the prepared baking tray.
Bake in the preheated oven until the chicken meat is no longer pink and the prosciutto is browned and crisp, about 40 minutes. Turn the chicken breasts over after 20 minutes.