Combine carrots, milk, and sugar in a 6- to 8-qt. Dutch oven; bring to a boil over high heat, stirring occasionally.

Reduce heat to medium; cook, stirring occasionally, until liquid has evaporated, about 1 hour.

Use a mortar and pestle or a rolling pin to break open cardamom pods. Coarsely crush seeds, discarding the shells.

Heat oil in a very large skillet over medium-high heat. Cook half of the cardamom until fragrant, about 30 seconds.

Add carrot mixture and cook, stirring constantly, until thick and jammy, about 15 minutes. Remove from heat.