Pat the chicken dry thoroughly with paper towels, then season generously with salt and pepper on both sides. Heat the olive oil in a 10-inch or larger skillet over medium-high heat until it starts to shimmer. If needed, work in batches and add the chicken to the pan. Sear the chicken until deeply browned on the bottom, about 6 to 7 minutes. Flip the chicken with tongs and sear the other side for another 6 to 7 minutes, until browned. Once done, transfer the chicken to a plate and set aside.

Lower the heat to medium, and add the butter to the skillet. Once melted, toss in the shallot, garlic, 1/2 teaspoon salt, and 1/4 teaspoon pepper. Stir occasionally and cook for about 1 minute, until the shallots start to soften. Add the flour, stirring to coat the shallots and garlic evenly, and cook for an additional minute.

Pour in the half-and-half or milk and chicken broth, stirring until the flour dissolves completely and any browned bits from the bottom of the pan are scraped up. Bring the mixture to a boil, then reduce the heat to a simmer. Return the chicken to the pan and let the sauce simmer for 3 to 4 minutes, until it thickens and coats the back of a spoon, with the chicken fully cooked through. Stir the lemon juice into the sauce, then garnish with chopped parsley and lemon slices. Serve immediately.