Place the chopped white chocolate in a food processor and pulse it until it’s in small crumbles.
Combine the milk, cream, and salt in a medium saucepan, and place the pan over medium heat. Bring the mixture to a simmer. With the food processor running, pour the hot cream into the processor and blend until the white chocolate is melted and the mixture is smooth.
Let the mixture cool to room temperature, then stir in the peppermint flavoring and a drop or two of green food coloring. Taste it, and add more mint flavoring if desired.
Pour the popsicle mixture into your popsicle molds, leaving about ¾-inch at the top. Fill each mold with a generous pinch of chopped cookies, enough to bring the popsicle mixture almost to the top of the mold. Insert a stick into each mold, using the stick to gently press some of the cookies down into the molds. Freeze for at least 4 hours.
To remove, run warm water over the popsicle molds for a few seconds and gently slide the pops out. If you encounter resistance, run a little more warm water over it until they slowly slide out.