Combine the first 3 ingredients in a medium bowl.
Peel bananas; cut each in half lengthwise. Cut each half into 4 pieces. Brush bananas with juice; dredge in brown sugar mixture.
Melt butter in a large nonstick skillet over medium-high heat; add bananas. Cook 2 minutes, turning bananas after 1 minute. Remove from heat; pour rum into one side of pan. Ignite rum with a long match; let flames die down.
Spoon 1/2 cup vanilla frozen yogurt into each of 4 dessert dishes; top each with 4 banana pieces and 2 tablespoons Pineapple Curd. Serve immediately.