Place the peeled and sliced carrots in a saucepan with just enough water to cover the bottom (about 1/4 cup) or use a steamer basket. Cover and cook over medium heat until the carrots are very tender when pierced with a fork, about 15-20 minutes.

Drain any excess water from the cooked carrots. Transfer them to a blender or food processor, adding 1/4 cup of fresh water if needed to help blend. Process until completely smooth. You should have about 1.5 cups of puree. Set aside to cool.

In a medium saucepan, combine the heavy cream, whole milk, granulated sugar, ground cinnamon, ground nutmeg, and salt. Heat over medium-low heat, stirring occasionally, until the mixture is hot and steaming, but not boiling.

While the cream mixture heats, whisk the egg yolks in a medium bowl until light yellow and slightly thickened. Gradually drizzle about half of the hot cream mixture into the egg yolks, whisking constantly to temper them. This prevents the yolks from scrambling.

Pour the tempered egg yolk mixture back into the saucepan with the remaining cream mixture. Continue to cook over medium-low heat, stirring constantly with a spatula or wooden spoon, until the custard thickens slightly and coats the back of the spoon, reaching about 175-180°F (80-82°C). Do not let it boil.

Remove the custard from the heat and stir in the vanilla extract and the cooled carrot puree. For an extra smooth ice cream, you can press the mixture through a fine-mesh sieve at this point to remove any fibrous bits. Cover the mixture and refrigerate for at least 4 hours, or preferably overnight, until thoroughly chilled.

Once the base is completely chilled, pour it into your ice cream maker and churn according to the manufacturer's instructions, typically 20-30 minutes, until it reaches a soft-serve consistency.

Transfer the soft ice cream to an airtight freezer-safe container. Press a piece of parchment paper or plastic wrap directly onto the surface to prevent ice crystals, then cover tightly. Freeze for at least 4-6 hours, or until firm. Let sit at room temperature for 5-10 minutes before scooping for the best texture.