Make the caramel: Combine the sugar and 1/4 cup water in a medium saucepan over medium heat, stirring just until the sugar dissolves. Cook, gently swirling the pan but not stirring, until amber, about 10 minutes. Brush any sugar crystals off the side of the pan with a wet pastry brush. Remove from the heat and carefully whisk in the butter, heavy cream, bourbon, lemon juice and kosher salt (the mixture will bubble). Pour into a bowl and set aside.
Make the bread pudding: Preheat the oven to 350 degrees F and butter a 9-by-13-inch baking dish. Spread the bread cubes on a rimmed baking sheet and bake until dry and golden, about 20 minutes. Let cool completely.
Whisk the eggs, sugar and kosher salt in a large bowl, then whisk in the milk, heavy cream and vanilla until smooth. Fold in the toasted bread and the banana slices. Transfer half of the mixture to the prepared baking dish and spread in an even layer. Drizzle with 1/2 cup of the caramel, then top with the remaining bread mixture. Cover with plastic wrap and refrigerate at least 1 hour or overnight. Cover and refrigerate the remaining caramel until ready to serve.
Preheat the oven to 350 degrees F. Sprinkle the bread pudding with sea salt and bake until golden brown and set, about 1 hour, 15 minutes. Warm the remaining caramel and drizzle on top.