1. Preheat oven to 350°.
2. To prepare strata, heat a large skillet over medium-high heat. Add 1 tablespoon oil and butter to pan; swirl until foamy. Stir in bananas; sauté 1 minute. Carefully add half of rum; cook rum until flames subside. Add remaining rum, 1 cup brown sugar, and 1/2 teaspoon salt; cook 3 minutes or until bananas are soft. Spoon the banana mixture into a large bowl, and press bananas with a pastry blender or fork until chunky. Combine half-and-half, granulated sugar, eggs, and evaporated milk in a bowl, stirring well with a whisk. Add banana mixture and bread cubes, tossing well to coat. Place bread mixture in a 13 x 9-inch metal baking pan coated with cooking spray. Bake at 350° for 20 minutes.
3. To prepare streusel, weigh or lightly spoon flour into a dry measuring cup; level with a knife. Combine flour, walnuts, and next 3 ingredients (through 1/8 teaspoon salt), stirring well with a whisk. Stir in 1 tablespoon oil. Remove strata from oven; sprinkle evenly with streusel. Bake an additional 30 minutes or until golden and set. Sprinkle with powdered sugar.