Toss onion with vinegar, sugar, and a pinch of salt in a medium bowl and let sit, squeezing onions occasionally to soften slightly and mellow flavor.

Make the aioli by whisking the mayonnaise, lemon herbs seasoning, gremolata oil and lemon juice in a medium bowl until well combined. Set aside.

Divide meat into 12 portions (about 3 oz. each). Place 1 portion on a work surface. Gently shape into ¾" thick patties. Season both sides with salt and pepper.

Heat mushroom sage olive oil in a cast iron pan or grill pan on medium-high heat. Grill patties until lightly charred on bottom, about 3 minutes. Flip and top with a cheese. Grill to desired doneness (cheese should be melted), about 3 minutes for medium-rare. Transfer patties to a cutting board and let rest 5 minutes.

While patties are resting, grill the buns, cut side down, until lightly charred around the edges, about 30 seconds.

Build sliders with patties, buns, onions, and aioli.