Lay the chicken breasts onto your work surface. Holding the breast down with your hand, use your very sharp knife to cut it in half horizontally. Repeat with the rest of the chicken, creating 4 pieces.

Brush both sides of each piece with mayonnaise and season with salt and pepper.

Use a meat mallet, rolling pin, or the bottom of a skillet to crush the crackers in the sleeve. Open the package and pour the crumbs into a shallow dish. There will be about 2 cups.

Dredge the chicken breasts in the cracker crumbs, generously coating both sides of the chicken. Transfer the coated breasts to a platter.

Heat the olive oil and butter in a sauté pan over medium-high heat. Once the butter is bubbling, add the chicken into the pan. (You can do this in batches if your pan will not accommodate all 4 breasts.) Cook until golden on one side, about 5 minutes. Turn and cook on the second side until golden, making sure that the chicken cooks completely, reaching an internal temperature of 165°, about another 5 minutes.

Transfer the chicken to a clean platter.

Add the flour to the leftover juices in the skillet. Reduce the heat to low. Whisk until a bubbly paste forms, about 2 minutes.

Whisk in the milk. Stir in the Parmesan cheese.

Season with salt and pepper.

Serve the sauce over the chicken. Garnish with fresh thyme.