Mix the butter and sugar together in a medium mixing bowl until smooth and combined. Add the flour, and then the vanilla and chocolate chips. Let the mixture sit for 10 minutes and then shape it into a disk, wrap in plastic wrap and refrigerate until firm, about 30-60 minutes.
MAKE ICE CREAM:
In a medium saucepan over medium-high heat melt the butter. Keep stirring and heating until the butter is deep golden brown in color and has a nutty aroma. Slowly whisk in the heavy cream and continue to heat until simmering. Then, remove the mixture from the heat and set aside.
Whisk together the brown sugar and the egg yolks in a medium bowl until pale in color. Add a small amount of the hot cream mixture to the bowl with the egg yolks, whisking constantly. Gradually whisk in the rest of the cream and the salt.
Return the egg-cream mixture to the medium saucepan and heat over medium-high heat until slightly thickened and the temperature reads 170-175 degrees F on an instant-read thermometer.
Remove from the heat and pour the mixture through a fine-mesh strainer into a large bowl. Whisk in the vanilla extract and the whole milk. Cover the bowl with plastic wrap and chill in the refrigerator for at least 4 hours or overnight.
Once chilled, freeze in an ice cream maker according to the manufacturer's instructions.
Cut the disk of cookie dough into small chunks.
Once the ice cream is done churning, transfer it to an air-tight ice cream container and stir in the chocolate chunks and the cookie dough chunks. Store in the freezer until firm, about 3-4 hours.