Slice the leeks in half, then thinly slice them into 1/4-inch-thick half-moons. Place the leeks in a colander set over a bowl and fill the bowl with enough water to cover the leeks. Gently stir them around to help remove any excess dirt and sand. Remove the colander with the leeks from the bowl, leaving behind any dirt. Let leeks drain completely.
Place the chicken breasts on a sheet of parchment paper. Season the chicken breasts with salt, a pinch red pepper flakes, and half the thyme leaves. Place another piece of parchment on top of the chicken. Use a meat mallet or a rolling pin to pound the chicken breasts slightly until they are an even thickness, about 3/4 to 1 inch thick. Hold both pieces of parchment paper and flip the chicken over; now the parchment that was originally on the bottom should be on top. Remove the parchment, exposing the unseasoned side of chicken. Season with salt, another pinch red pepper flakes, and the remaining thyme leaves. Place the parchment back on top and pound slightly to adhere the seasoning.
Heat a large straight-sided skillet over medium-high heat. Add in 2 tablespoons olive oil and swirl to coat the pan. Add in the chicken breasts and cook, undisturbed, about 4 minutes. Flip the chicken and cook until the chicken is cooked through and golden brown and reaches 165 degrees F, 4 to 5 minutes. Remove the chicken to a plate and set aside.
Turn the heat down to medium and add in the remaining 2 tablespoons olive oil. Add the fennel, mushrooms, and leeks to the pan and cook, stirring occasionally, until the vegetables have softened, 5 to 7 minutes. Season the vegetables with 1/4 teaspoon each salt and pepper. Deglaze with the white wine and cook until the wine is mostly evaporated, about 1 minute. Turn the heat down to low and add in the cream. Cook, stirring frequently, until the cream has thickened slightly, 2 to 3 minutes. Season the sauce to taste.
To serve, spoon the creamy vegetables over the chicken. Garnish with thyme leaves.