To make the cookie dough: Combine the butter, salt, and brown sugar together in a small bowl. Mix until smooth, breaking up any brown sugar clumps. Stir in the flour until incorporated, then add the vanilla and chocolate chips. Turn the mixture onto a piece of plastic wrap, shape into a disk then wrap and refrigerate until well chilled and firm. (If your butter was still hot, the mixture may be loose - just stick the bowl in the fridge for a few minutes and allow it to firm up before shaping and wrapping.)

To make the ice cream: Add the butter to a medium saucepan and set over medium heat. The butter will melt then begin to bubble and foam before eventually starting to brown - swirl the pan occasionally to help it brown evenly. Continue cooking until it becomes a dark shade of golden brown and smells nutty - be patient and watch carefully, it can go from brown to burned quickly. Add the heavy cream and whisk to combine. Allow the mixture to come to a simmer.

Meanwhile, whisk the brown sugar, egg yolks, and salt in a heatproof bowl until pale in color. Slowly add the warm cream mixture to the egg yolks, whisking constantly. Pour back into the saucepan and set over medium-low heat. Stir constantly until the mixture thickens enough to coat the back of a wooden spoon (about 170-175 F on an instant read thermometer). Strain the custard into a heatproof bowl and stir in the vanilla and whole milk.

Cover and refrigerate until thoroughly chilled (overnight is good). Freeze in your ice cream maker according to the manufacturer's instructions. Meanwhile, chop the disk of cookie dough into bite-size pieces. As you transfer the ice cream to an airtight container for storing, mix in the cookie dough pieces. Freeze until firm, at least 1 hour.