Put the brown sugar, butter, cinnamon, allspice, and salt in a large skillet over low heat. Cook, stirring occasionally, until the butter has melted and the mixture begins to bubble, about 15 minutes. Add the amaretto and cook, stirring, until the mixture is smooth, then raise the heat to medium and add the bananas. Cook until the bananas are coated in the sauce and warmed through, about 5 minutes.
Add the rum to the skillet. Light a match or lighter and lower it to the surface of the sauce until it ignites. When the flame subsides, turn off the heat. Divide the ice cream into four bowls, then divide the bananas and sauce among the bowls. Serve immediately.
