Cut each piece of bread into four triangles by slicing diagonally. Cut or break the chocolate into 8 pieces, roughly the same triangle shape as the bread pieces but slightly smaller by at least 1/4 inch all the way around.

In a medium bowl, vigorously whisk together the eggs and sugar until well combined. Add the half-and-half, vanilla, and salt and whisk until combined. Put the bread pieces in the bowl and press down gently to make sure they're all soaked.

Heat 1 tablespoon of the butter in a medium nonstick skillet over medium heat. When the butter is melted and sizzling, add half of the bread pieces in one layer, leaving a little space between each. Cook until nicely browned, about 2 minutes. Turn each piece over with a spatula and cook until the other side is nicely browned, 1-1/2 to 2 minutes. Transfer the bread pieces to a plate lined with paper towels and take the skillet off the heat. Put a triangle of chocolate on half of the bread pieces, and top each with another piece of bread (save the best-looking pieces for the tops). Let them sit while you return the pan to the heat, melt the remaining 1 tablespoon butter, cook the remaining bread, and fill it with the remaining chocolate. When all the chocolate sandwiches are made, arrange two, overlapping slightly, on each of four plates. Sprinkle the confectioners' sugar generously over all and serve warm.