Heat 1 tablespoon of olive oil in a large skillet over medium-high heat. Once the oil shimmers, indicating it's hot, add the chicken breasts. Cook for 4-5 minutes per side until golden brown and cooked through. Remove and set aside.

Reduce the heat to medium and add the remaining tablespoon of olive oil to the skillet. Add the flour and stir constantly for 1 minute, allowing it to cook and lose its raw taste.

Slowly pour in the chicken broth, whisking continuously to prevent lumps. The mixture should thicken slightly, resembling a smooth sauce. This step ensures a lump-free sauce with a consistent texture.

Add the lemon juice and capers to the skillet. Stir to combine and let the sauce simmer for 2-3 minutes. The aroma of lemon should be prominent, signaling the sauce's readiness.

Return the chicken breasts to the skillet, nestling them into the sauce. Spoon the sauce over the chicken, ensuring each piece is well-coated. This step allows the flavors to meld.

Cover the skillet with a lid and allow the chicken to gently simmer in the sauce over low heat for an additional 5 minutes. This slow cooking process helps the chicken to stay tender and moist, while it absorbs the sauce's rich aroma and flavors, enhancing the overall dish.

Use an instant-read thermometer to check the internal temperature of the chicken breasts. Ensure it reaches 165 F (74 C) to confirm they are cooked safely. If the temperature is lower, continue cooking for a few more minutes, checking periodically to avoid overcooking.

After the chicken breasts reach an internal temperature of 165°F (74°C) and are golden brown, transfer them from the skillet to a serving platter. Let them rest for about five minutes to retain their juices while you complete the sauce preparation in the skillet.

If the sauce becomes overly thick, gradually incorporate a small amount of chicken broth, stirring continuously over medium heat. This will help achieve a smooth, pourable consistency. Adjust the thickness to your preference, ensuring the sauce coats the back of a spoon without being too runny or too dense.

Before serving, sprinkle the chicken with freshly chopped parsley. This final touch not only enhances the visual appeal with its bright green hue but also introduces a gentle herbal aroma. The parsley's subtle flavor provides a refreshing contrast to the savory elements, ensuring a well-rounded dish presentation.