In a mixing bowl, beat butter with electric mixer on low speed until creamy, then blend in the egg. Slowly add dry sugar cookie mix and blend until all incorporated. Gather dough and divide evenly into six balls, then dye each ball a color of the rainbow using the natural food coloring.
Stack dough balls on top of each other beginning with red, then following with orange, yellow, green, blue and purple. Flatten dough balls with your hand to make one large disc of dough, then divide dough in half and and form into flat rectangles, mixing the colors together. Wrap in plastic and chill 15 minutes until firm.
Remove one half of dough and roll out evenly until the dough is about 1/4" thick. Using a knife, trim the rounded ends of the rectangle so the edges are straight, then cut the dough evenly into 8 rectangles and transfer to a baking sheet lined with parchment paper. Repeat with the other half of the dough.
Preheat the oven to 375 degrees. Place one teaspoon of strawberry jam into the center of half the rectangles. Place another rectangle of dough on top of the jam and seal each side closed with a fork. Place on the baking sheet and freeze 10 minutes. Bake 20 minutes, then transfer to a wire rack to cool completely.
To make the glaze, whisk together the milk and dry glaze mix in a bowl until smooth and creamy. Drizzle over cooled hand pies.