Butter six 10-ounce ramekins, large muffin pan, or mini cast iron skillets
Combine water, cinnamon sticks, and piloncillo in a small saucepan. Bring to a boil; reduce heat, creating a syrup. Simmer syrup uncovered for 20 minutes. Remove from heat and let steep, covered for 1 hour. Discard cinnamon sticks.
Preheat oven to 350 degrees.
In a large bowl, whisk together syrup, eggs, milk, bourbon, vanilla, cinnamon, and salt. Toss in the bread cubes, and stir gently to evenly coat; let stand a few minutes. Divide among prepared dishes, pressing down slightly to make level.
Bake until set in the center and top is golden, about 40 minutes. If bread browns too quickly, cover loosely with aluminum foil. Remove from oven; let cool slightly.
Bourbon Sauce:
Melt butter in a medium saucepan over medium heat. Remove from heat and add sugar, milk, bourbon, and salt, stirring until incorporated and smooth.
Mixed Berry Sauce:
Combine the berries, sugar, and cornstarch in a saucepan and stir until the sugar is dissolved. Continue to simmer it until the berries have released all of their liquid, about 5 to 8 minutes. The sauce will thicken slightly. Use a wooden spoon to combine.
Assembly:
To serve, un-mold bread pudding onto plates.
Pour bourbon sauce over top of bread pudding and top with warm berry sauce and a scoop of vanilla ice cream.
Notes
If you don't have any day-old bread on hand, bake fresh bread cubes in the oven for about 10 minutes. You can also swap in croissants for an even butterier take. YUM!
Bread pudding is best served warm, but can be kept refrigerated, tightly covered, for up to 4 days.