Heat 1 tablespoon olive oil in a large skillet over high heat. Season the pork with salt and pepper, then brown on all sides, about 5 minutes. Transfer to a plate. Reduce the heat to medium low; add the remaining 1 tablespoon olive oil, the shallots and carrot to the skillet and cook until the vegetables are just tender, about 5 minutes.
Add the sage and wine and bring to a simmer. Return the pork to the skillet along with the tuna and capers. Cover and cook until a thermometer inserted into the pork registers 150, 10 to 15 minutes. Transfer the pork to a cutting board and let rest about 5 minutes.
Meanwhile, discard the sage and transfer the remaining contents of the skillet to a blender. Add the reserved tuna oil, the mayonnaise, lemon juice, and salt and pepper to taste; puree until smooth.
Slice the pork and divide among plates. Top with the sauce and greens, season with salt and pepper, and sprinkle with more capers. Per serving: Calories 485; Fat 30 g (Saturated 5 g); Cholesterol 103 mg; Sodium 744 mg; Carbohydrate 12 g; Fiber 2 g; Protein 36 g