In a saucepan, simmer the milk, half of the rum, brandy, and cinnamon sticks for 5 minutes, being careful not to boil.

Toss the sliced bananas with the ground cinnamon and lemon juice in a bowl.

Melt the butter in a skillet over medium-high heat. Add the bananas and cook until softened only a few minutes. Remove from the cook top and in a safe work area, add the remaining rum and flambe with caution.

When the flame dies a moment later, carefully transfer the bananas and cooking liquid to the saucepan of milk. Simmer for 10 minutes longer.

Strain through a fine-mesh sieve before dividing and serving.

Sprinkle with a dash of cinnamon if desired.