Make the tuna sauce: Puree the tuna, anchovies, mayonnaise, 2 tablespoons olive oil, the lemon juice and 2 tablespoons water in a blender until smooth.
Place each turkey cutlet between 2 sheets of plastic wrap and pound with a meat mallet or heavy skillet until 1/8 to 1/4 inch thick. Brush with the remaining 1 tablespoon olive oil and season with salt and pepper.
Heat a grill pan over medium-high heat and lightly brush with olive oil. Grill the turkey, in batches if necessary, until just cooked through, about 2 minutes per side.
Pile the arugula, tomatoes and capers in the center of the turkey cutlets. Drizzle with half of the tuna sauce and roll up to enclose. Drizzle with the remaining sauce and sprinkle with the basil.