Place 2 brioche slices on a work surface and top each with 3 slices of cheese in a single layer. Spread ½-ounce caviar on top of the cheese and cover with a slice of salmon, trimmed to fit the size of the brioche. Top each with a second slice of brioche.
Melt 1-tablespoon butter in a nonstick skillet over medium heat and place the sandwiches in the pan. Cook, pressing down lightly with a spatula, until golden brown, about 3 minutes. Add the remaining tablespoon of butter to the pan and flip the sandwiches, cooking until golden and the cheese has melted, about 2 minutes.
Transfer the sandwiches to a cutting board, slice diagonally, and serve immediately.