Heat milk and fresh ginger in a medium saucepan over medium-low heat until steaming but not simmering, 3 to 5 minutes. Turn off heat, cover, and steep for at least 1 hour. Crumble goat cheese in a medium bowl and set aside.
Whisk egg yolks, sugar, and salt in a medium bowl for about 1 minute, until mixture is silky and light yellow.
Strain milk (discarding ginger) and return pot to medium-low heat, stirring occasionally, until steaming and warm but not simmering, 3 to 5 minutes.
Whisk about ⅓ of the warm milk into the egg mixture until combined. Add egg mixture to the remaining warm milk and stir to combine. Cook, stirring constantly, for 5 to 7 minutes, until custard thickens and coats a spatula.
Strain custard through a sieve into the bowl with the goat cheese. Whisk mixture until goat cheese melts and custard is smooth. Transfer to a lidded container, cover, and refrigerate at least 4 hours or overnight.
Pour chilled custard into the chilled bowl of your ice cream maker and process according to the manufacturer’s directions. About halfway through churning, add candied ginger. Finish churning, then transfer to a lidded container, cover, and freeze for at least 2 hours or overnight. Soften at room temperature for about 5 minutes before serving.