Step 1

For the ice cream: Preheat the oven to 400 degrees. Line a baking sheet with parchment paper.

Step 2

Toss the carrots with the oil and 1 tablespoon of the honey. Arrange them in a single layer on the prepared baking sheet. Roast (middle rack) for 20 to 35 minutes, turning once halfway through, until tender and just starting to caramelize (see headnote). Let cool, then transfer to a food processor and puree until smooth. The yield is 1 cup.

Step 3

Whisk the egg yolks in a medium bowl, until smooth.

Step 4

Combine 3 cups of the cream, the sugar, vanilla bean pod and seeds, and the remaining 3 tablespoons of honey in a medium saucepan over medium-low heat. Whisk in the carrot puree until completely incorporated. Cook, whisking, until the mixture reaches 130 degrees, then remove the pan from the heat.

Step 5

While whisking, slowly drizzle about one-quarter of the cream-carrot mixture into the egg yolks (to temper it), then pour it all back into the pan and stir with a wooden spoon to combine. Cook over medium-low heat, stirring continuously, until the mixture reaches between 170 and 175 degrees. Do not go above this range or the egg yolks may scramble. Immediately remove from the heat and stir in the remaining 1 cup of cream. Strain the mixture through a fine-mesh strainer into a glass container, cover and refrigerate overnight.

Step 6

Churn the chilled ice cream base in an ice cream maker according to the manufacturer's directions. Scrape the ice cream into a jar or pan, cover with a piece of plastic wrap or parchment paper pressed directly to the surface of the ice cream, and freeze until firm, about 3 hours.

Step 7

For the crumble: Preheat the oven to 350 degrees. Line a rimmed baking sheet with parchment paper.

Step 8

Combine both sugars, the butter and water in a small saucepan over medium heat. Cook, stirring continuously, until the sugars have dissolved and the mixture is bubbling. Stir in the nuts and sesame seeds, then remove from the heat and pour onto the baking sheet. Use a spatula to spread the nuts and seeds into an even layer. Bake (middle rack) for about 13 minutes, until the syrup has set and is lightly golden. Let cool completely.

Step 9

Even though the cooled slab may still be pliable, break it into pieces and transfer to a food processor; pulse to a coarse-crumb consistency.

Step 10

To serve, scoop the ice cream into bowls and top with a sprinkling of the crumble and pink salt, if using.