1.Store butter and eggs at room temperature.
2.Preheat oven to 180°C (160°C fan-forced). Grease deep 20cm (8-inch) square cake pan; line base and sides with baking paper.
3.Cut three thin slices from the centre of peeled pineapple (reserve remaining pineapple for another use). Halve and core each slice. Heat grill pan over high heat; grill pineapple about 2 minutes each side until charred and starting to soften.
4.Halve vanilla bean, scrape half the seeds into medium saucepan; add caster sugar and the water. Stir over high heat, without boiling, until sugar dissolves. Bring to the boil; boil, without stirring, until a light caramel colour. Remove from heat; stir in the extra butter and half the rum. Pour into cake pan; place pineapple around edge of pan.
5.Beat butter, brown sugar and remaining vanilla seeds in small bowl with electric mixer until light and fluffy. Beat in eggs, one at a time. Transfer mixture to large bowl. Stir in remaining rum, then sifted flour and milk, in two batches.
6.Spread mixture over pineapple; bake cake about 1 hour.
7.Stand cake in pan 5 minutes before turning onto serving plate. Serve cake warm with vanilla ice-cream and extra finely chopped pineapple.
Cake will keep in an airtight container at room temperature for 1 day, or in the refrigerator for 2 days.Note