In a small saucepan, over medium heat, melt the butter. Cook the butter until it is light brown and smells nutty, stirring often. Be careful not to let it burn.
Add the half and half to the butter and bring just to a simmer.
Meanwhile, in a medium bowl, whisk brown sugar, egg yolks, and salt for a minute or two, until light and fluffy. Slowly pour a cup of the warm half and half into the egg mixture, whisking while you pour (this will slowly bring up the temperature of the eggs). Pour the mixture back into the saucepan. Heat, stirring often, until the mixture slightly thickens and reaches 170 degrees or until it coats the back of a spoon.
Remove from the heat and stir in the vanilla extract. Allow the mixture to cool slightly. Then transfer to a bowl, cover, and refrigerate until cold.
Once the mixture is chilled, whisk in the 1 1/2 cups of whole milk. Pour into an ice cream maker and freeze accordingly to manufacturer's directions.
While the ice cream is churning, immediately prepare the cookie dough.