Bananas: Butter the inside of a slow cooker, line completely with foil, and then butter the foil. Turn on to HIGH. Sprinkle butter, brown sugar and rum over the foil on the bottom of the slow cooker. Cover the bottom with the banana halves, cut side down, in a slightly overlapping pattern. Press the bananas into the sugar
Cake: Sift the flour, baking powder, cinnamon, nutmeg, and salt into a large bowl; then whisk to combine evenly.
In another bowl, slowly beat the butter and sugar with an electric mixer until just blended. Raise the speed to high and beat until light and fluffy, scraping the sides of the bowl occasionally, about 10 minutes. Beat in the egg and then the yolk, allowing each to be fully incorporated before adding the next.
While mixing slowly, add the flour mixture to the butter in 3 parts, alternating with the milk in 2 parts, beginning and ending with the flour. Mix briefly at medium speed to make a smooth batter.
Pour the batter over the bananas and smooth with a spatula to even it out. Lay a doubled length of paper towel from end to end over the top of the slow cooker, to line the lid and create a tighter seal. (See note.) Cover the cake tightly with the lid and continue to cook on HIGH, until the cake begins to brown slightly on the sides and springs back when touched in the middle, about 3 1/2 hours. Turn off the slow cooker and let the cake set, about 20 minutes more.
Using the foil, lift the cake from the slow cooker; set on the counter to cool, about 30 minutes more. Fold back the foil, and carefully invert cake onto a platter so you can see the caramelized bananas.
Slice or spoon cake into bowls and serve with ice cream, if desired.