Step 1: Make the caramelized shallots. Melt the butter with the oil in a cast iron skillet or heavy saucepan. Add the shallots. Cook over medium-low heat until the shallots have been reduced to a dark, thick, sweet paste. This will take about 30 minutes and you’ll have to stir often. Lower the heat as needed to keep the shallots from burning. Add salt and pepper to taste. The shallots can be caramelized several hours or even a day ahead.
Step 2: Make the burgers. Divide the beef into 24 portions. Lightly wet your hands with cold water and mold each into a small patty about 2 inches across and 1/2-inch-thick, or if you have one, use a special slider mold. Arrange on a plate lined with plastic wrap and refrigerate, covered with more plastic wrap, until you’re ready to grill.
Step 3: Set up your grill for direct grilling and preheat to high. In the best of all worlds, you’d grill over oak or other hardwood embers. Brush and oil the grill grate.
Step 4: Generously salt and pepper the sliders and arrange on the grate. Grill until cooked to taste, 3 to 4 minutes per side for medium, turning with a spatula. Meanwhile, brush the cut sides of the rolls with melted butter and grill them until golden brown as well.
Step 5: To assemble the sliders, place a spoonful of caramelized shallots on the bottom section of each bun. Top with a slider, pickle slice, and the top bun, and serve at once.