Mise en Place: Halve the tomatoes and olives; thinly slice the shallot and mince the garlic. Juice the lemon. Measure out the wine, broth, and seasonings. Pat the chicken breasts thoroughly dry with paper towels.
Season the Chicken: In a small bowl, combine the salt, pepper, rosemary, thyme, and red pepper flakes. Using a knife, carefully make two shallow, parallel scores across the top of each chicken breast. Sprinkle the herb mixture evenly over all sides of the chicken.
Sear the Chicken: Place the Dutch oven over medium-high heat and add the olive oil. Once shimmering, carefully place the chicken breasts in the pan in a single layer, ensuring not to crowd them. Sear for 3-4 minutes per side, until a deep golden-brown crust forms. Transfer the seared chicken to a plate.
Build the Sauce Foundation: Reduce the heat to medium. Add the sliced shallot to the pan and cook for 2 minutes, stirring to scrape up any browned bits. Add the minced garlic and cook for another 30 seconds until fragrant.
Deglaze and Simmer: Pour in the white wine to deglaze the pan, stirring to release any remaining flavorful bits from the bottom. Allow the wine to reduce by about half, which should take 2-3 minutes. Stir in the chicken broth and fresh lemon juice.
Braise to Perfection: Return the seared chicken to the Dutch oven. Scatter the halved tomatoes and olives into the liquid around the chicken. Bring the sauce to a gentle bubble, then reduce the heat to low, cover the pot, and cook for 12-15 minutes. The chicken is done when it is tender and an instant-read thermometer inserted into the thickest part registers 165°F (74°C).
Garnish and Serve: Remove the Dutch oven from the heat. Garnish generously with fresh parsley and optional crumbled feta. Serve the chicken immediately with a spoonful of the pan sauce and tomatoes.