Place ¼ cup flour in a shallow bowl. Cut chicken breasts in half horizontally and pound until they're ½-inch thick. Season both sides with salt and pepper and dredge in flour, shaking off the excess.
Heat olive oil in a large saute pan over medium heat. Lay chicken in one even layer in the pan and cook until golden brown. Flip and cook on the other side until golden brown. Remove and set aside.
In the same pan, melt butter. Add garlic and saute for 30 seconds. Add remaining 1 tablespoon flour, paprika, and red chili flakes, stirring until well combined.
Add sherry and chicken broth and bring to a simmer. Reduce heat to low and simmer for 2-3 minutes or until sauce is slightly thickened. Add lemon juice and stir to combine. Adjust the seasoning the sauce with salt and pepper.
Add chicken back to the pan and spoon the sauce on the chicken. Let simmer for 2 minutes or until chicken is hot. Remove from heat and sprinkle parsley on top. Serve immediately.
Notes
If preparing this dish ahead of time, you may need to add a splash of water to help loosen the sauce. The sauce will thicken slightly as it cools.