Put the condensed milk, coconut milk, butter, corn syrup, and 1/2 cup of the coconut in a 3-quart heavy-duty saucepan and bring to a boil over medium heat. Turn the heat to medium low and cook, whisking constantly, until the mixture thickens and pulls together into a dense batter, about 8 minutes. When the mixture is ready, the whisk will leave trails in the batter, allowing you to briefly see the pan bottom, and when you tilt the pan, the mixture should slide to the side in a blob, leaving a thick residue on the bottom of the pan. (It's OK if the residue is slightly brown.)

Slide the mixture into a bowl. (Don't scrape the pan--you don't want to use any of the batter stuck to the bottom.) Let the mixture cool to room temperature and then refrigerate until very firm, 3 to 4 hours.

Put the remaining 1/2 cup coconut in a bowl. Using a teaspoon or a melon baller, scoop the mixture by the teaspoonful, and with your hands, roll each into a ball about 1 inch in diameter. Drop each ball into the coconut as you finish rolling it. When you have 4 to 6 brigadeiros, roll them in the coconut and lift them out with your fingers semi-open, carefully shaking off the excess.

Store brigadeiros in a tightly covered container at room temperature for up to two days or in the refrigerator for up to two weeks. (If refrigerating, bring to room temperature before serving for the best flavor and texture.)

If you can't find finely shredded unsweetened coconut, you can buy coconut chips or flaked coconut and run it through a food processor until it looks like it was grated on the smallest holes of a box grater. Be sure it's unsweetened, or the brigadeiros will be much too sweet.