Have the brewed coffee thoroughly chilled before starting; place two tall glasses in the freezer to chill and set the ice cream out for 5 minutes to soften slightly.
Add the cold milk to a blender, then pour in the chilled coffee.
Spoon in the vanilla ice cream.
Blend on medium-high until smooth and thick, 20–30 seconds, stopping once to scrape down if needed; the shake should flow in a steady vortex and look creamy.
Adjust thickness to taste: add 1–2 tablespoons more milk for a looser shake or a little more ice cream for extra thickness, then pulse 5 seconds.
Pour into the chilled glasses and top with whipped cream (for serving). Serve immediately.