Make sure your ice cream bowl is frozen before starting the recipe.
For the brown sugar ice cream:
Combine 2 cups milk and 2 cups heavy creamin a medium saucepan and bring to a simmer. Remove from the heat.
Meanwhile, put 1 cup brown sugar in a clean saucepan larger than the one with the milk, and melt over medium heat. It will be grainy at first and will start to come together and deepen in color. Be careful not to burn it. When it's a deep brown, which will be a few minutes since it's already brown, remove from heat and add the milk mixture. It will bubble up furiously. Be careful. Stir with a wooden spoon or spatula with a long handle.
Return to low heat and stir until the caramel is completely melted.
In a large bowl, whisk 6 egg yolks to mix.
Very slowly add the hot caramel mixture, just a bit at a time at first, so you don't curdle the eggs. Whisk constantly while adding the rest of the milk mixture and return to the same pan where you made the caramel.
Cook it over low heat for about 5 minutes, until it starts to thicken. Be careful not to let it boil.
Strain through a fine mesh to catch any curdled bits and transfer to a clean bowl. Add ¼ teaspoon vanilla extract and pinch sea salt if using. Let it cool to room temperature. Cover and refrigerate until very cold if needed for your ice cream machine (it might take several hours).
Churn in an ice cream maker according to the manufacturer's directions.
Transfer to a metal or freezer-safe container and freeze for several hours until solid.
Notes
Chilling the custard before churning: this depends on the ice cream machine you use. The less expensive ones made for home use usually need a very cold mixture, so I recommend making it the day before. Better machines can handle room-temperature or hotter mixtures, so read the instructions before. Serving: eat it on its own, caramelized fruit (cooked in butter and brown sugar until golden but not too soft), with a berry sauce or chocolate fudge sauce.