Make a tea of the cinnamon by placing the water and cinnamon sticks in a 1 quart saucepan and bringing to a boil for 3 minutes. Remove the tea from the heat and allow to cool completely.
Cut the bread into 1-inch by 1-inch chunks. Place in a baking pan, and toast under the broiler for 3 to 5 minutes.
Every minute or so, remove the bread pan from the broiler and toss the bread so that it toasts evenly. Remove the bread from the broiler.
In a large 12-inch skillet, heat one stick of butter until melted and very slightly brown. Add the pecans and allow to sauté for about 45 seconds.
Using a slotted spoon, remove the pecans and place in a large mixing bowl. Keep your pan at a medium low heat.
Add half of the toasted bread chunks to the pan, toss to coat with the butter and then remove, placing them in the large mixing bowl.
Add half of the remaining stick of butter, melt, and then add the remaining bread chunks, coating with butter and removing to the bowl as before.
Once all of the bread is in the bowl, add the apricots. Beat the eggs well and then pour into the mixing bowl with the bread and other ingredients.
Toss all of the ingredients well so that the mixture is well coated with the beaten eggs.
Preheat oven to 350°F.
Coat the inside of an ovenproof 2 quart casserole by rubbing the remaining half stick of butter on the inside surface. (Not all of the butter will be used in this step.) Pour the ingredients from the large bowl into the buttered casserole.
Add what is left of the last half stick of butter to the skillet plus brown sugar and 2 cups of the cinnamon tea. Combine well and allow to simmer for 1 minute.
Remove from the heat and pour over the contents of the casserole. Cover the casserole with foil and place in the heated oven. Bake for 1 hour. Remove from the oven and allow to cool for 5 minutes.
Immediately before serving, heat the dulce de leche in a mixing bowl for 45 seconds in the microwave.
Remove from the microwave and whisk in rum, bourbon or orange juice. Serve with the budín.