Combine milk and cream in medium saucepan. Whisk in combined sugar, dextrose, dry milk and stabilizer. Cook on medium heat, stirring occasionally, until mixture reaches 160°F. Remove from heat.
Strain mixture into blender and process 30 seconds. Transfer to clean bowl and chill over an ice bath. Refrigerate overnight.
Place a quarter sheet pan in the freezer. Freeze milk mixture in ice cream maker according to manufacturer's directions. Fold in cookie pieces. Line frozen pan with parchment paper. Spread ice cream evenly in pan. Place variegate in pastry bag fitted with #3 plain tip. Swirl variegate through ice cream. Smooth top and freeze until solid.
Cut 12 (2-1/2-inch) circles from frozen ice cream. Keep frozen until ready to assemble.
Flatten each donut to about 2-1/2-inches in diameter. Place on parchment paper-lined sheet pan. Cover with a second pan and add weights so donuts remain flat.
When ready to assemble, heat panini grill. Place 1 ice cream circle between 2 donuts. Grill just until donuts are warm. (Timing will vary.) Serve immediately.