Place flour in a small bowl or plate. Sprinkle chicken breasts evenly on both sides with salt, pepper, and about 1 teaspoon of oregano. Warm a drizzle of olive oil in a large skillet over medium-high heat.
Lightly dredge each chicken breast through the flour. Shake gently to remove any excess, then place each piece in the hot pan. Cook for 2-3 minutes on each side, just until browned and cooked through. Remove chicken to a plate and set aside.
Add another small drizzle of olive oil to the skillet, along with the butter. When butter melts, add the garlic and cook for 30-60 seconds, just until fragrant. Pour in the chicken broth, then scrape the pan to remove any browned bits from the bottom. Bring the liquid to a simmer and cook for 2-3 minutes, until slightly reduced.
Stir in the cream, followed by the remaining 1/2 teaspoon oregano and salt and pepper to taste. (I usually add 1/4 teaspoon salt and 6-8 cranks of black pepper; this will vary according to your family’s preference.) When cream is warmed through, stir in the lemon juice.
Return chicken to the skillet. Let it warm for 1-2 minutes, spooning a bit of sauce over each piece.
Garnish as desired with extra lemon wedges, grated lemon zest, and fresh herbs, such as parsley or dill. Serve and enjoy!