For the Ice Cream
Arrange banana slices in single layer on foil-lined baking sheet. Freeze until solid, 2 hours.
Put 1/2 frozen slices into food processor bowl fitted with metal blade. Add 1/2 lemon juice, 1/2 sugar and 1/2 rum. Process until thick and almost smooth. With processor on, add 1/2 milk through feed tube. Process until smooth, about 10 seconds. Transfer to 3-cup airtight container and reserve. Repeat with other half bananas, lemon juice, sugar, rum and milk. Put reserved mixture back into bowl. Process 5 seconds. Transfer to same container and freeze 1 hour. (Can be frozen up to 3 days.) Return mixture to food processor and process until smooth. Before serving, let sit at room temperature 10 minutes.
For the Caramel Sauce
Place brown sugar, syrup, butter, cream and dash salt in medium nonstick skillet over medium high heat. Cook, stirring, until boiling and thickened, with large bubbles breaking on surface, about 4 minutes. Remove from heat; stir in rum to taste. (Can be made several days ahead and refrigerated. Sauce becomes very thick; gently reheat in microwave oven or on stove until just warm and liquid.)
To serve, divide banana slices among 6 shallow soup plates. Put 3 small scoops Ice Cream on each. Drizzle with warm sauce. Serve immediately, passing remaining warm sauce separately.