Using a mixer, cream the sugar and butter until fluffy, scraping the sides once. Add the eggs one at a time, mixing after each addition. Mix in the vanilla.
Sift together the flour, cocoa, baking soda and salt in a bowl, then add in batches to the mixer, mixing until just combined after each addition. Gently mix in the semisweet, milk and white chocolate chips.
Using a cookie scoop, add generous 1-tablespoon portions of dough to a baking sheet (about 12 portions). Repeat with the remaining dough and additional baking sheets. Dot the top of each cookie with 2 to 3 chocolate chips if you'd like them to really show up on top. Bake until the cookies are barely done and are still soft and chewy, 9 to 11 minutes. Don't overbake! Let cool on a baking rack.
Melt the semisweet chocolate in a bowl over a pan of simmering water.
Dip the cookies one by one into the melted chocolate to coat, then transfer to parchment-lined baking sheets to set.