Preheat the oven to 350 and line the cookie sheet with parchment paper. Grease lightly.
In a mixer, beat butter and sugar until light and smooth (a couple of minutes).
Beat in the egg and scrape the sides of the mixer.
Add vanilla and milk, mix until just combined.
On low speed, add cocoa powder, flour, baking soda and salt. Mix on low speed until just combined. Scrape the sides of the mixer.
Add the food coloring and mix until just evenly incorporated, scrape the sides and mix just until all even color.
Fold in the M&Ms with a spatula.
Roll the cookie dough into 1-inch balls and place them on a cookie sheet about 2 inches apart.
Bake for 8-10 minutes. (8 if you want them softer and chewier.)
Cool the cookies completely on a cooling rack.
Add about 1/4 cup of cheesecake ice cream between two cool cookies and press very lightly and carefully, so not to break the cookie. Place the cookies in a zip-lock freezer bag, get some of the air out and lay them flat in the freezer. Freeze for about an hour or two before eating. (If you eat them right away, before freezing, it will be a big mess with ice cream falling out.)