Heat oven to 350 degrees F. Coat a 9-inch x 13-inch baking dish with non-stick cooking spray.
Place bread pieces in a large mixing bowl; set aside.
In a large skillet over medium-high heat, melt butter and 1 tablespoon oil together. Add banana slices, and cook fro 1 minute.
Carefully add half the rum. Cook for 1 to 2 minutes (or until flames subside if rum flames). Add the rest of the rum, brown sugar, and 1/2 teaspoon salt. Cook until sauce reduces down and thickens a bit, and sugar is completely dissolved, about 3 to 4 minutes.
In a medium mixing bowl, beat together eggs, evaporated milk, half-n-half, and sugar. Pour the milk mixture over the bread. Then pour the banana mixture over the bread. Gently stir to coat the bread and combine everything together. {The mixture is going to look very liquidy, that's okay.}
Carefully pour the pudding mixture into the prepared baking dish. Bake for 20 minutes.
Streusel:
In a medium bowl, combine flour, pecans, brown sugar, oats, and salt. Add oil, and stir until evenly moistened (the mixture will still be very dry).
Remove the pudding from the oven and sprinkle the streusel evenly over the top of the pudding. Return pudding to the oven and bake for an additional 30 minutes, or until top is golden and pudding i set.
Remove from oven and let cool at least 10 minutes before serving. Sprinkle the entire pudding or individual servings with powdered sugar if desired.