Preheat the oven to 350°F.
In a large stockpot, place the potatoes and a generous pinch of salt. Cover with cold water and bring to a simmer. Cook at a simmer until the potatoes are fork-tender, about 15 minutes. Drain the potatoes and return to the pot. Add the butter, sour cream, heavy cream, cream cheese, salt and pepper. Mash until smooth.
To make the sauce, heat the olive oil in a medium saucepan. Add the garlic and cook for 1 minute, until fragrant. Add the chile flakes and cook for 1 to 2 minutes. Stir in the crushed tomatoes and bring to a simmer. Stir in the basil and remove from the heat. Season with salt and pepper.
In a bowl, combine the roasted tomatoes, parmesan and mozzarella.
Slice the chicken breasts in half lengthwise and gently pound to flatten. Place the chicken in a baking dish with sides and fill each piece of chicken with about 1-2 tablespoons of the tomato-cheese mixture. Roll the chicken breasts up and place seam side down in the baking dish.
Place the breadcrumbs in a shallow bowl and stir in the garlic powder, grated parmesan, lemon zest, parsley and a big pinch of salt. Place the eggs in a second dish and the flour in a third. Season each with salt and pepper.
Dip the stuffed chicken first into the flour, then into the eggs, and lastly into the breadcrumbs and place in the baking dish seam side down. Repeat with all of the chicken.
Bake the chicken for 10 minutes. Pour in the tomato sauce, sprinkle with shredded mozzarella and parmesan and bake for another 5 minutes, until cooked through and melted.
Meanwhile, bring a large pot of salted water to a boil for the beans. Prepare an ice bath by filling a large bowl with cold water and ice. When the water boils, add the green and yellow beans for 2-3 minutes then immediately transfer to the ice bath to stop the cooking process. Remove from the ice bath after 1-2 minutes and return to the empty pot. Toss with olive oil, garlic and season with salt.
Divide the stuffed chicken between plates, cover in sauce and serve with beans and mashed potatoes on the side.