Heat the sugar and water in a large heavy skillet over medium heat until starts to bubble. Stop stirring and continue to cook, swirling the skillet periodically. Continue this process (without stirring) until it is deep amber in color, about another 8 to 12 minutes. Remove from the heat.
Meanwhile, warm 1¼ cup of the heavy cream.
Add the butter and the warmed cream into the sugar mixture, and stir constantly until smooth (it will bubble vigorously for a minute, but will calm down).
Transfer to a bowl and stir in the salt and vanilla.
Chill until no longer warm, about 15 to 20 minutes.
Meanwhile, bring milk, the remaining cup of cream, and the maple syrup to a boil in a small heavy saucepan, stirring occasionally.
In a medium bowl, whisk the eggs and then add half of the hot milk mixture in a slow stream, whisking constantly.
Pour all of the egg and milk mixture back into the 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). If it starts to scramble, take off the heat and stir continuously until smooth and thickened.
Pour custard through a fine-mesh sieve into a large bowl, then stir in cooled caramel.
Refrigerate the custard for 3 hours, or overnight.
Freeze custard in an ice cream maker (it will be soft), then transfer to an airtight container and place in freezer until firm, 4 hours, or overnight.
Serve with toffee bits sprinkled on top, if desired.