Place the chicken breasts onto your work surface. Use a sharp chef’s knife to butterfly the breasts by cutting them in half horizontally to create 4 uniform, thin cutlets.

Brush each piece thoroughly on both sides with the Dijon mustard, providing a tacky surface for the breading, and season with salt and pepper.

Place the crackers inside the sleeve and use a meat mallet or rolling pin to crush them into small, irregular shards.

Pour the cracker crumbs into a shallow dish. Dredge each mustard-coated chicken breast in the crumbs, pressing firmly to ensure a generous, architectural coating on both sides.

Heat the olive oil and butter in a large sauté pan over medium-high heat. Once the butter is bubbling, place the chicken into the pan.

Cook until the crust is deeply golden, approximately 5 minutes. Carefully turn the cutlets and cook on the second side until the internal temperature is reached and the crust is crisp, another 5 minutes. Transfer to a clean platter.

Reduce the heat to low. Sprinkle the flour over the remaining fats in the skillet, whisking for 2 minutes to form a bubbly roux paste.

Slowly whisk in the milk to initiate starch gelatinization. Once thickened, stir in the grated Parmesan until the sauce is a smooth, stable emulsion.

Taste and adjust seasoning with salt and pepper as needed.

Plate the chicken and drape with the warm Parmesan cheese sauce. Garnish with fresh thyme sprigs and serve immediately.