Heat the sugar in a large heavy skillet over medium heat, stirring with a wooden spoon to heat sugar evenly, until it starts to melt, then stop stirring and cook, swirling skillet periodically so sugar melts evenly, until it is amber in color.
Meanwhile, warm 1 1/4 cup of the heavy cream.
Carefully pour the warmed cream into the melted sugar, and stir constantly until all of the caramel has dissolved.
Transfer to a bowl and stir in the salt and vanilla.
Cool to room temp.
Meanwhile, bring milk, remaining cup of cream, and the maple syrup to a boil in a small heavy saucepan, stirring occasionally.
Lightly whisk eggs in a medium bowl, then add half of the hot milk mixture in a slow stream, whisking constantly.
Pour back into saucepan and cook over medium heat, stirring constantly with a wooden spoon, until custard coats the back of the spoon, 4 - 5 minutes. (Don't let come to a boil, it should reach 170° F on an instant-read thermometer).
Pour custard through a fine-mesh sieve into a large bowl, then stir in cooled caramel.
Refrigerate the custard for 3 - 6 hours.
Freeze custard in ice cream maker (it will be soft), then transfer to an airtight container and place in freezer until firm, 3 - 4 hours, or overnight.