Trim crusts and roll each bread slice flat with a rolling pin until half its original thickness.

Stir softened butter, brown sugar, cinnamon, and salt into a smooth paste and spread thinly on each slice, leaving a quarter inch bare along one long edge.

Roll each slice tightly from the coated edge and rest seam side down for 3 minutes.

Whisk eggs, milk, vanilla, and cinnamon until completely smooth with no white streaks.

Dip each roll for 2 seconds, turning once, and let excess drip off.

Cook 4 at a time in foaming butter over medium heat, seam side down for 90 seconds, then turning to brown all sides, about 5 minutes total.

Roll the hot rolls in cinnamon sugar immediately, then rest on a wire rack.

Beat cream cheese smooth, add powdered sugar, vanilla, and milk to a ribbon consistency, and drizzle over the warm roll ups.