Combine the goat cheese with the corn syrup in a mixing bowl.
Heat the cream and milk, stirring continuously, in a saucepan until nearly boiling and small bubbles form at the edge of the bowl, 5 to 10 minutes.
Meanwhile, whisk the egg yolks and sugar into a separate mixing bowl until pale yellow, 4 to 5 minutes. Add a quarter of the hot cream mixture to the egg mixture to temper it. Then add the egg mixture to the cream mixture in the saucepan. Heat the mixture over low heat, whisking continuously, until the mixture coats the back of a spoon, about 10 minutes.
Pass through a fine mesh strainer into a bowl, and then place the bowl into an ice bath until cool, at least 30 minutes or overnight. Add the goat cheese mixture and stir to combine, refrigerate until cooled completely. Add to an ice cream maker and prepare according to the manufacturer's instructions.