Light a grill. In a mini food processor, pulse the cornichons with the parsley leaves, capers and garlic clove until finely chopped. Add the mayonnaise and mustard and pulse until blended.
In a large bowl, gently mix the ground beef with the ground lamb and season generously with salt and pepper. Form the meat mixture into four 1-inch-thick patties and make a slight indentation in the center of each one with your thumb. Season the burgers with salt and pepper.
Grill the English muffin halves cut side down until toasted, about 1 minute. Flip the muffins and grill for 30 seconds longer. Transfer to a work surface cut side up and spread the bottom halves of the muffins with the caper remoulade.
Grill the burgers over high heat until they are nicely charred on the bottom, about 4 minutes. Flip the burgers, top them with the cheddar cheese and grill until cooked to medium and the cheese melts, about 4 minutes longer. Transfer the burgers to the toasted English muffins and let rest for 5 minutes. Top the burgers with the tomatoes and cucumbers if desired, close the burgers and serve.