Melt 1 tablespoon butter in a small skillet over medium-low heat. Add sliced onion and cook slowly, stirring occasionally, until deep golden and jammy, about 12 minutes. If they threaten to burn, add a splash of water. Set aside.

While onions cook, gently warm chopped brisket in a covered dish in the microwave or a low oven until just heated through, about 2 minutes. You want it warm, not hot enough to steam the bread later.

Spread Dijon on one side of each bread slice. Divide half the cheddar between two slices, top with warmed brisket, caramelized onions, and remaining cheddar. Cap with remaining bread, mustard side down. Spread remaining 2 tablespoons butter on the outside of both sandwiches.

Heat a large skillet over medium-low heat. Cook sandwiches until bread is deeply golden and cheese is fully melted, about 4-5 minutes per side. Press gently with a spatula for even contact. If cheese is slow to melt, cover briefly with a lid.

Transfer to a cutting board, let rest 1 minute so the cheese sets slightly, then slice diagonally. Serve with pickle juice on the side for dipping, or drizzle a few drops over the cut sandwich for brightness.