For the sliders: Heat the olive oil in a medium skillet over medium heat. Add the onions, season with a bit of salt and pepper, and cook until softened and translucent, about 5 minutes. Let cool.
In a large bowl, combine the sauteed onions, five-spice powder and 1 tablespoon of the balsamic vinegar, and season with a generous pinch of salt and of pepper. Add about one-quarter of the ground beef, and gently mix together until combined. Add the remaining ground beef and gently mix, making sure not to overmix.
Divide the meat into approximately 2-ounce portions, and form each into a ball. Insert a cube of butter into the center of each ball, then lightly press into patties about 3/4-inch thick and 2 inches in diameter. Place the patties on a baking sheet, cover with plastic, and refrigerate until ready to cook.
Fire up the grill or grill pan to medium-high. Cook the sliders about 2 to 3 minutes per side for medium doneness. After flipping, drizzle each slider with a bit of the remaining balsamic. Set two pieces of cheddar on each slider about 2 minutes before it is done cooking. Place the sliders on buns, and serve with sauces on the side.
For the sweet Russian sauce: Combine the mayonnaise, cornichons, ketchup, balsamic and capers in a small bowl; mix until blended.
For the arugula pesto: Combine the garlic and 2 tablespoons of the olive oil in a mini chopper or food processor, and pulse until the garlic is chopped. Add the toasted walnuts and pulse until finely chopped. Add the arugula, the remaining 2 tablespoons olive oil and the lemon juice, and season with salt. Pulse just until everything is finely chopped, not pureed.