In a large bowl, combine the warm milk and active dry yeast, stir once, and let sit for 5 minutes until foamy.
Whisk in the granulated sugar, whole egg, egg yolk, melted butter, and 1 tsp vanilla bean paste until smooth.
Add 2 cups of the all-purpose flour and the salt to the bowl and mix until a shaggy dough forms.
Add the remaining 1 cup flour and knead by hand or with a dough hook until the dough is smooth and pulls away from the sides of the bowl.
Add blue gel food coloring a little at a time and continue kneading until the color is evenly distributed and the dough is soft and elastic.
Place the dough in a lightly greased bowl, cover with plastic wrap or a clean towel, and let rise in a warm place for 1 to 1 1/2 hours, until doubled in size.
While the dough rises, make the filling by mixing 1/2 cup softened butter, brown sugar, and ground cinnamon in a medium bowl until well combined.
Stir the 1/2 cup crushed chocolate sandwich cookies and 1/4 cup mini chocolate chips into the cinnamon-sugar mixture until evenly distributed.
Lightly flour a work surface, punch down the risen dough, and roll it out into a 14 by 9 inch rectangle with the long side facing you.
Spread the cookie-cinnamon filling evenly over the dough, leaving a small border along the far long edge.
Starting from the long edge closest to you, roll the dough up tightly into a log, pinching the seam to seal.
Slice the log into 12 equal rolls using a sharp serrated knife or unflavored dental floss.
Grease a 9 by 13 inch baking dish and arrange the rolls cut side up in the dish, leaving a little space between each roll.
Cover the pan and let the rolls rise in a warm place for 30 to 45 minutes, until puffy.
Preheat the oven to 350°F while the rolls complete their second rise.
Bake the rolls for 20 to 25 minutes, until the tops are lightly golden and the centers are just set.
While the rolls bake, make the icing by beating the cream cheese and 1/4 cup softened butter together in a bowl until smooth and creamy.
Add the powdered sugar and 1/2 tsp vanilla bean paste to the bowl and beat until combined, then add the milk and mix until the icing is smooth and spreadable.
Remove the rolls from the oven and let them cool for 10 minutes, then spread or drizzle the cream cheese icing over the warm rolls.
Sprinkle the 1/2 cup crushed chocolate sandwich cookies and additional mini chocolate chips over the iced rolls before serving.