1. Combine the brown sugar, water and salt in a saucepan and bring to a boil, stirring to dissolve the sugar. Boil the syrup for 2 minutes.
2. Meanwhile, beat the eggs together in a medium bowl. Slowly beat in the syrup.
3. Cook in a double boiler over, not in, boiling water, stirring constantly, until the mixture reaches 175°F and coats the back of a spoon. Do not allow the mixture to boil. Add the unsalted butter and stir until melted. Strain into a medium bowl and refrigerate until cold.
4. Add the whole milk and vanilla extract and mix well to combine.
5. Beat the heavy cream until soft peaks form. Fold it into the egg mixture.
6. Pour into an ice cream maker and freeze according to the manufacturer's directions. Fold in the chopped pecans before transferring ice cram to a container.