Prepare the pudding. Whisk eggs in a large mixing bowl to beat lightly. Whisk in milk, 1 cup white sugar, 1 cup brown sugar, heavy cream, butter, vanilla extract, and salt until evenly combined.

Tear sourdough and brioche breads into bite-sized pieces. Add both breads to the custard mixture and stir with a rubber spatula until well coated. Allow to stand at room temperature and soak for 20 to 30 minutes, stirring occasionally.

Meanwhile, preheat the oven to 325 degrees F (165 degrees C). Grease a 9x13-inch baking dish with butter.

After soaking, spoon bread mixture into the prepared baking dish.

Bake in the preheated oven for 55 minutes.

Meanwhile, whisk together the remaining 2 tablespoons white sugar, remaining 2 tablespoons brown sugar, and cinnamon in a small bowl.

Remove the pudding from the oven and sprinkle the cinnamon-sugar mixture evenly over the top. Increase oven temperature to 450 degrees F (230 degrees C).

Return pudding to the oven and bake until firm with a lightly crisped top, about 6 minutes. Cool at least 20 minutes before slicing.

While pudding cools, prepare the sauce. Beat egg in a small mixing bowl. Whisk in the heavy cream.

Combine sugar and butter in a medium saucepan over medium heat. Cook, stirring frequently, until butter melts and sugar dissolves. Remove saucepan from heat. While rapidly whisking the butter and sugar mixture, slowly pour in the beaten egg and cream mixture, whisking until entirely incorporated. Whisk in bourbon and return to medium heat. Cook, whisking constantly, for 1 to 2 minutes. Whisk in salt.

Cut the pudding into squares and serve with warm bourbon sauce.