Combine mango, lime juice, and lime zest in the bowl of a food processor. Process until pureed. Add eggs, salt, and nutmeg; process 15 seconds longer.

Combine sugar and butter in a bowl. Add to mango mixture in the food processor. Process until smooth.

Transfer mixture to a microwave-safe bowl. Heat in the microwave at 50% power for 2 minutes; stir with a whisk. Heat in the microwave for 2 minutes more; whisk again. Continue heating and whisking until the mixture achieves a pudding-like consistency, 6 to 10 minutes total. Cover and refrigerate until cold, about 30 minutes.