Lightly butter an 8-inch square pan.

Melt ghee in a large skillet set over medium-low heat. Add carrots and cook, stirring, for 8 minutes, until they begin to soften and lose some of their moisture. Slowly pour in milk, then sprinkle on coconut.
Raise the heat to medium and cook, stirring occasionally, until carrots and coconut have absorbed milk and the mixture has the consistency of rice pudding, 45 to 55 minutes. It’s OK if the milk boils occasionally, but do remember to stir throughout the process to prevent sticking.

Stir in sugar and cook until the mixture has thickened and clings to itself in a soft ball when stirred, about 8 minutes. Stir in cardamom and remove from heat.

Pour carrot mixture into the prepared pan and use a silicone spatula to make the layer even. Once the pan cools to room temperature, refrigerate until completely cool and cut into small squares.
Burfi will keep, refrigerated, up to 5 days.