Heat a griddle pan over medium heat. Place a baking sheet in oven and heat to 275°F.
Preheat a cast-iron skillet for the ham steak over medium-high heat.
Whisk up the eggs, half-n-half, extracts and nutmeg. Soak the bread in the custard on each side. Butter the griddle and cook 2-3 slices at a time cooking until they are deep golden on each side and the custard has cooked through the bread. Keep toast warm in oven and repeat with the remaining bread slices.
When the toast is done, heat a second skillet over medium-high heat and melt 3 tablespoons butter. When it foams, add the bananas and cinnamon stick; sprinkle a little freshly grated nutmeg and the dark brown sugar overtop. Toss to melt, add the maple syrup and heat through; remove pan from heat, add the rum and ignite. When the flame subsides, place the pan back on the stove and keep warm over low heat.
Add a drizzle of oil to the skillet then add the ham steak, brown on each side and heat through. Douse pan with a little water to deglaze. Carve ham steak into 4 portions.
Layer 1 slice of toast, a slice of ham and another slice of toast. Top with ¼ of the bananas and serve.