Pat the chicken breasts dry with paper towels. If they are very thick, carefully slice them horizontally to create two thinner cutlets, or pound them lightly to an even thickness of about 0.75 inch. Season both sides generously with salt and pepper.

Heat the olive oil in a large skillet over medium-high heat until shimmering. Add the seasoned chicken breasts and cook for 5-7 minutes per side, or until golden brown and cooked through with an internal temperature of 165°F (74°C). Remove the chicken from the skillet and set aside on a plate, tented loosely with foil to keep warm.

Reduce the heat to medium. Add the butter to the same skillet, letting it melt. Add the minced garlic and cook for 30-60 seconds, stirring constantly, until fragrant but not browned.

Pour in the chicken broth, scraping up any browned bits from the bottom of the pan with a wooden spoon. Bring the broth to a gentle simmer and cook for 2-3 minutes, allowing it to reduce slightly.

Whisk in the heavy cream, fresh lemon juice, and Dijon mustard. Continue to simmer gently for 3-5 minutes, stirring occasionally, until the sauce has thickened slightly and coats the back of a spoon.

Remove the skillet from the heat. Stir in the fresh parsley and dill. Taste and adjust seasoning with additional salt, pepper, or lemon juice if needed.

Return the cooked chicken breasts to the skillet, spooning the creamy lemon herb sauce over them. Garnish with additional fresh parsley, if desired. Serve immediately with your favorite sides like rice, pasta, or roasted vegetables.