For the chicken: Put an oven rack in the center of the oven. Preheat the oven to 150 degrees F. Line a baking sheet with a wire rack.
Using 3 wide shallow bowls, add the flour to 1, the eggs to another and to the third bowl combine the bread crumbs, Parmesan cheese, basil, and thyme.
On a work surface, put the chicken between 2 pieces of plastic wrap. Using a meat mallet, lightly pound the chicken until approximately 1/4 to 1/2-inch thick. Season the chicken with salt and pepper. Dredge the chicken pieces in the flour to coat lightly, then dip into the beaten eggs, allowing the excess egg to drip off. Coat the chicken with the bread crumb mixture, pressing gently to adhere.
In a large, nonstick saute pan, heat the vegetable oil over medium heat. Add 2 pieces of the breaded chicken into the oil and cook until light golden brown, about 3 to 4 minutes on each side. Transfer the chicken to the prepared baking sheet and keep warm in the oven. Repeat with the remaining chicken. Reserve the cooking juices in the saute pan.
For the sauce: Using the same saute pan, add the olive oil to the reserved cooking juices and heat over medium heat. Add the fennel and cook, stirring frequently, until softened, about 5 to 6 minutes. Stir in the cherry tomatoes, 1/4 teaspoon salt, 1/4 teaspoon pepper, garlic and thyme. Cook for 5 to 6 minutes until the tomatoes are tender. Remove the pan from the heat. Add the mascarpone cheese and stir until the mixture is creamy. Season with salt and pepper, to taste.
Transfer the chicken to a serving platter and spoon the sauce over the top before serving.