Preheat the broiler. Place the hatch chiles on a small rimmed baking sheet. Broil for 8 to 10 minutes, turning the chiles occasionally, until charred on all sides. Remove from the oven and transfer to a resealeable bag. Let rest for about 20 minutes, until cool enough to handle. Peel the chiles, remove the seeds, and dice.
Preheat the oven to 350°F. Beat the cream cheese, ricotta, and goat cheese in the bowl of an electric mixer on medium speed for 2 to 3 minutes, until light and fluffy. Add the egg, sugar, and salt and blend until creamy. Stir in the diced chiles.
Pour the mixture into a 4½-inch springform pan. Place the pan in a larger baking pan. Add enough hot water to the baking pan to come halfway up the sides of the springform pan. Bake for about 45 minutes, until the cheesecake is set around the edges and jiggles only slightly when gently shaken. Remove from the oven and chill in the refrigerator for 3 to 4 hours, until cold. Serve with crackers for dipping.