Combine whiskey and cookie pieces in a jar; shake well. Let stand overnight. Strain through a coffee filter, gently pushing all liquid from cookies. Discard solids. Set aside until ready to use.
Ice Cream: Meanwhile, combine milk and cream in medium saucepan. Whisk in combined sugar, dextrose, milk powder and stabilizer. Cook on medium heat, stirring occasionally, until mixture reaches 165°F. Stir in 3 oz. cookie pieces. Let stand at room temperature 30 min. Strain; discard solids.
Blend ice cream mixture, in small batches, in blender 30 seconds. Pour into medium bowl. Chill bowl over ice water bath until mixture is cold. Stir in vanilla. Refrigerate overnight.
Freeze mixture in ice cream maker according to manufacturer's directions. When firm, stir in remaining 3 oz. cookie crumbs. Transfer to clean container. Freeze until ready to use.
For each serving: Pour 2 oz. infused whiskey and 12 oz. of root beer in a chilled tall glass. Top with one #24 scoop (2.5 oz.) ice cream. Garnish glass with 3 whole cookies on side of glass.