Slice each piece of chicken into 2 cutlets by placing your hand on top of the chicken breast and running a sharp knife horizontally through the breast, slicing the top and bottom into two thin, even pieces.
In a bowl or container large enough to fit a chicken breast, stir together flour and salt. Dredge each chicken breast piece in the mixture until lightly coated.
In a 12-inch skillet with a lid, heat oil over medium-high heat. Once hot, add chicken breasts and cook until well-browned, 2-3 minutes. Flip and repeat with the second side. Transfer chicken breasts to a plate.
Turn heat down to medium and add the butter to the pan. Once the butter is melted, add garlic and optional red pepper flakes and cook until garlic is lightly golden, about 1 minute.
Add honey, water, soy sauce, vinegar, and cornstarch/water mixture and bring to a simmer, using a wooden spoon to scrape any cooked-on bits from the bottom of the pan. Carefully add chicken back to the pan. Cover and cook, stirring occasionally, until chicken is cooked through, 5 to 8 minutes. If at any point the sauce becomes too thick and looks like it might burn, add a couple tablespoons of water to the pan.
Transfer chicken breasts to a clean plate and allow them to rest for 5 minutes. If sauce is not quite thick enough, you can allow it to continue simmering uncovered while your chicken rests until it reaches your desired consistency.
Serve the chicken topped with the sauce and a side of rice.