Fill a large pot 3/4-full with water. Place the unopened can of condensed milk in the water, making sure it is covered completely, and carefully bring to a gentle boil. Reduce heat and simmer for 3 hours, adding more water as necessary. Remove from the heat and let sit until cool enough to handle. Watch the can carefully to make sure it does not start to bulge. If the can does begin to bulge, remove from the heat and let cool. Once cool, punch a small hole in the top of the can, return to the heat, and continue cooking.
If you are uncomfortable using the whole can method, you can cook the condensed milk, out of the can, over a double boiler, stirring every 5 minutes, for about 3 to 4 hours, or until it has achieved a deep golden color.
In a clean saucepan, combine the brown sugar and 1/4 cup water. Bring to a boil over medium heat. Reduce the heat and simmer, stirring, until the sugar is dissolved. Remove from the heat and let cool.
In a metal bowl set over a pot of simmering water, whisk together the warm syrup and egg yolks. Whisk constantly until the mixture is a thick, pale yellow and reaches the ribbon stage. Remove the bowl from the heat, and beat with an electric mixer on low speed until cool. Whisk in the condensed milk, heavy cream, vanilla and salt. Strain through a fine mesh sieve into a clean container. Cover with plastic, pressing down onto the surface to prevent a skin from forming. Refrigerate until well chilled, at least 2 hours, preferably overnight.
Transfer to the bowl of an ice cream machine and process according to the manufacturer's instructions. Transfer to an airtight container and freeze until ready to serve.
Yield: generous 1 quart