In a large bowl, combine 2 cups cookies with milk. Let soak for at least 30 minutes in the fridge. Transfer to a blender and blend on high until smooth. Let sit for 10 to 15 minutes. Remove the foam that develops with a spoon and discard. Set aside.

In a medium saucepan, whisk together the sugar, cornstarch, and salt. Slowly whisk in the infused milk and egg yolks. Place over medium heat and cook, stirring constantly, until bubbling. Lower heat and cook for about 1 minute, continuing to whisk, until thickened. Remove from the heat, transfer to a large measuring cup with a spout, through a fine mesh sieve and stir in the vanilla extract.

Fill a popsicle mold halfway with the pudding, add a few crushed cookies into the middle of each popsicle and then continue to fill them to the top. Give the mold a few taps on the counter to dislodge any air bubbles. Cover with the lid, insert the popsicle sticks and freeze until solid for at least 4 hours or overnight. Can be kept in the freezer for up to 3 days. To remove from mold, run warm water on the bottom of the mold for about 1 minute to loosen popsicles.