Preheat oven to 425°F. Line a rimmed baking sheet with foil. Spread carrots on pan. Drizzle with butter and sprinkle with sugar and ¼ teaspoon salt. Roast, stirring once, until very tender, about 30 minutes.

Reduce oven temperature to 300°F. Arrange six 8-ounce ramekins in a large, shallow baking or roasting pan.

Add roasted carrots and milk to a blender container. Cover and blend until very smooth, about 1 minute.

Whisk together the cream, brown sugar, cinnamon, cardamom, nutmeg, and remaining ⅛ teaspoon salt in a medium saucepan. Heat over medium, stirring regularly, until small bubbles form around the edges or until an instant-read thermometer reaches 170°F, 3 to 5 minutes. Whisk in puréed carrot mixture and vanilla bean paste.

Beat the egg yolks in a medium bowl until very smooth, about 1 minute. Slowly whisk in the hot carrot mixture.

Strain the mixture through a fine mesh sieve into a glass measuring cup. Divide the mixture evenly between the ramekins. Fill the baking pan with water to halfway up the sides of the ramekins.

Bake, uncovered, until set and the center still wobbles, about 45 to 50 minutes. For a more firm custard, bake for a full 60 minutes.

Remove the ramekins from the water bath and transfer to a wire rack to cool. Cover with plastic wrap and refrigerate 2 hours, or until chilled, or up to 5 days.

Combine sugar with carrot juice in a small saucepan. Cook, swirling occasionally but not stirring, over medium-high heat until mixture turns a golden brown and reaches 200°F, about 4 minutes.

Serve the pot de crème with whipped cream and the carrot caramel.