Melt the butter in a small saucepan over medium heat. Swirl the pan often as it foams, and watch for tiny brown specks to form at the bottom. Once it smells nutty and toasty, pull it off the heat right away it goes from perfect to burnt fast.
Whisk the brown sugar straight into the warm browned butter until it dissolves into a thick, glossy mixture. Slowly stream in the heavy cream while whisking constantly so it doesn't seize up. Add the vanilla, cinnamon, and salt, and stir until it's silky and smooth.
While the syrup cools slightly, pull your espresso shots or brew a strong batch of coffee. I like using espresso because it holds up against the richness of the syrup, but strong drip coffee works in a pinch too.
Heat your milk until it's warm but not boiling, then froth it using a handheld frother or a whisk until it's light and foamy. You want soft, pillowy microfoam, not a stiff meringue-like foam.
Spoon 1½ to 2 tablespoons of the brown butter syrup into your mug, then pour in the hot espresso and stir until fully combined. Top it off with your steamed milk, spoon a little foam on top, and enjoy while it's warm.