For the meat: On a plate, brush the beef with the oil. Sprinkle with the green peppercorns. Season generously with salt. Let sit out at room temperature for 30 minutes.
For the green peppercorn mayonnaise: Meanwhile, in a small bowl, combine all of the ingredients. Cover and refrigerate until ready to serve.
For the skewers: In another bowl, combine all of the ingredients. Season with salt and pepper.
Thread the mushrooms alternating with the onion onto skewers. Set the remaining marinade aside for cooking.
Preheat the grill, setting the burners to medium-high. Oil the grate.
Grill the mushroom skewers for 10 minutes, turning them over a few times during cooking. Brush with the reserved marinade towards the end of the cooking time.
At the same time, grill the beef for 3 to 4 minutes on each side for medium-rare, ensuring the internal temperature reaches 63°C (145°F). Set aside on another plate.