Tear the bread into small pieces. In a small bowl, combine the bread, milk, beaten eggs, and sugar. Stir well until bread pieces are moistened.
Add the red food coloring and stir until the color is evenly distributed.
Cover and refrigerate for 15 to 30 minutes to soften the bread and absorb more liquid.
Transfer the mixture to a medium pan and over medium heat, cook the pudding while continually stirring to prevent from sticking to the bottom of the pan.
When most of the liquid is absorbed, add the butter and stir until melted and distributed.
Continue to cook the pudding into a chunky, paste-like consistency. Turn off the heat and stir in the vanilla extract.
Let the mixture cool completely before using.
In a large mixing bowl, combine warm water, active dry yeast, and 2 teaspoons of the sugar. Let the mixture sit for about 5 minutes or until foamy.
Add the warm milk, the melted butter, the remaining sugar, and salt. Stir everything well with a wooden spoon.
Add 2 1/4 cups of flour, about one cup at a time, stirring well after each addition. Gradually add and stir in the remaining flour, just until the dough gathers up into a shaggy mass.
Turn the dough over on a board and knead until it turns from loose and shaggy to smooth and elastic.
Place the kneaded dough in a bowl and cover with a clean kitchen towel. Let rise for 1 1/2 hours, or until doubled in size.
Gently deflate the risen dough and transfer to a flat board. Roll out the dough into a 10 x 16-inch rectangle using a rolling pin and slice in half horizontally.
Spoon a heaping mound of the pudding in the center length of each half. Using a spatula, slightly flatten the filling.
Fold the edges of the dough towards the middle to cover the pudding and pinch the seams to close.
Turn the logs around to seam side down and slice the logs to 2-inches pieces.
Arrange on a parchment-lined baking sheet and cover the pan loosely with a clean kitchen towel or plastic wrap. Let rise for about 30 to 40 minutes or until puffy.
In a small bowl, beat the egg with the 1 tablespoon water and brush the tops of the dough with the egg wash.
Bake in a preheated 350 F oven for 15 to 18 minutes, or until golden.