In a bowl, mix the softened butter, light brown sugar, and cinnamon until well combined. Trim crusts off the bread slices and flatten each slice gently with a rolling pin.
Spread the cinnamon butter mixture over each bread slice, then roll them up tightly.
In a shallow dish, whisk eggs, milk, and vanilla extract until smooth. Dip each roll-up into the egg mixture, ensuring full coverage.
In a skillet, melt the remaining butter over medium heat. Cook roll-ups until golden on all sides, turning as needed.
Mix granulated sugar and cinnamon in a small bowl. Roll each cooked roll-up in the cinnamon-sugar mixture.
For icing, beat softened cream cheese and confectioners’ sugar until smooth. Drizzle over warm roll-ups or serve on the side.