Preheat the oven to 350 degrees F. Line a baking sheet with parchment paper or a silpat mat (or spray with nonstick spray).
In a large bowl, use an electric mixer to combine the butter and sugars until well mixed. Mix in the egg and vanilla.
In a separate bowl, whisk together the flour, malt, cocoa, baking soda, cream of tartar, and salt. Add the dry ingredients to the wet and mix until the flour is incorporated. Stir in 1 1/4 cups of chocolate chips.
Using a cookie scoop or a tablespoon, scoop about 2 tablespoons of dough onto the cookie sheet in a rounded lump. Dot 5 or 6 additional chocolate chips on top of each cookie dough lump, if desired. Space cookies 3-inches apart on the cookie sheet (8 per cookie sheet works well).
Bake cookies for 12 to 14 minutes, or just until the cookies are almost set and no longer super soft in the middles. Let them cool on the baking sheet for a few minutes before transferring them to a rack to cool completely.
*Malt powder can usually be found near the hot chocolate in your market.