In a saucepan, combine heavy cream, whole milk, brown sugar, ground cinnamon, ground ginger, ground nutmeg, and salt. Heat over medium-low until warm and sugar dissolves, about 5 minutes.

Whisk the egg yolks in a separate bowl. Gradually add about ½ cup of the warm mixture into the egg yolks while whisking constantly.

Pour the egg mixture back into the saucepan, stirring continuously. Cook over low heat, stirring, until thick enough to coat the back of a spoon, about 10 minutes.

Remove from heat and stir in vanilla extract and finely grated carrots. Blend until creamy using an immersion blender.

Transfer the mixture to a container, cover with plastic wrap, and refrigerate for at least 4 hours or overnight.

Churn in an ice cream maker according to manufacturer's instructions until it reaches soft-serve consistency, about 20-25 minutes, adding mix-ins in the last 5 minutes.

Transfer the ice cream to a freezer-safe container, cover, and freeze for at least 4 hours before serving.