Bring a medium pot of water to a boil while you dice potatoes. Add potatoes to boiling water and salt water liberally. Boil potatoes until tender, 12 to15 minutes.
While potatoes cook, prepare burgers. Combine pork and veal in a bowl with 3 tablespoons extra-virgin olive oil, 1/4 cup (eyeball it) red wine, onion, sage, garlic, salt and pepper then form 4 large patties.
Preheat a large nonstick skillet over medium high heat. Place the burgers into the skillet, leaving a space in the center of the pan to pile in mushrooms. Add sliced mushrooms to the skillet with burgers and add 2 tablespoons olive oil. Flip burgers after 6 minutes and toss the mushrooms around as they brown at the center of the skillet. After the mushrooms begin to brown, season them with salt and pepper. The color will be deeper and the mushrooms will remain firmer if you wait for them to brown before salting. Cook burgers 5 minutes on opposite side then remove them from the pan to roll bottoms on a serving plate. Place sliced cheese on burgers, then the hot mushrooms. Cover plate loosely with foil to slightly melt cheese. Add remaining 1/4 cup of wine to skillet and loosen drippings. Dip roll tops in pan drippings to soak them up. Pile arugula on each burger then set roll tops in place.
Drain potatoes and return them to the warm pot to dry them out. Take the pot over to your cutting board and add in capers, celery, onion, lemon zest and juice, red wine vinegar and 3 or 4 tablespoons of extra-virgin olive oil to pot. Toss to combine the salad, then season the salad with salt and pepper, to your taste. Transfer salad to a serving dish. Potato salad can be served warm or cold.