Bring 1½ cups water, sugar, and vanilla to a boil in a saucepan over high heat. Add 3 or 4 pear halves, reduce heat to medium and simmer gently until the pears have just turned tender, about 5 minutes. Remove cooked pears to a serving dish, and continue to cook the remaining pears.
Increase heat to medium-high, and boil the syrup until it has reduced to 1 cup. Stir in the apricot jam and return to a boil. Dissolve the cornflour in 2 tablespoons of water and stir into the simmering syrup. Cook and stir until thickened and clear, about 30 seconds.
To serve, pour the hot sauce over the pears and sprinkle with the rum. Carefully ignite the rum tableside with the lights turned low. Let the alcohol burn out before serving.