Begin by seasoning the chicken cutlets generously on both sides with salt and pepper. Set them aside for the time being.
In a large bowl, whisk together the mayonnaise and ¼ cup of chimichurri sauce. Reserve the remaining chimichurri sauce. Add the seasoned chicken cutlets to the mixture, ensuring they are evenly coated. You can cook them immediately or for enhanced flavor, transfer them to a sealed container and refrigerate for 4 to 24 hours.
If grilling, preheat a gas or charcoal grill over high heat for about 10 minutes. Cook the chicken cutlets directly over the high heat, turning and flipping them occasionally, until they are just cooked through and exhibit a light char on all sides, which should take approximately 4 to 5 minutes. Once done, transfer the chicken to a serving platter. Spoon some of the reserved chimichurri sauce over the chicken and serve the remainder in a small bowl on the side.
For skillet cooking: Heat a large (12-inch) cast-iron or nonstick skillet over medium-high heat until a drop of water immediately balls up and moves rapidly across the surface. Add the chicken cutlets in a single layer and cook, swirling and flipping them occasionally, until they are evenly browned and just cooked through, which typically takes about 4 minutes. Transfer the cooked chicken to a serving platter. Spoon some of the reserved chimichurri sauce over the chicken and serve the rest in a small bowl on the side.