1. Set a sieve over a bowl and line it with two layers of cheesecloth. Pour yogurt into sieve, cover with plastic wrap, and refrigerate for 6 to 8 hours to drain. Discard the liquid and measure out ¾ cup strained yogurt. Add buttermilk and set aside. Place canister for ice cream maker inside the freezer to chill.
2. Start the frozen yogurt: In a small bowl, mix 2 tablespoons of the milk and the cornstarch to make a smooth slurry. In a medium bowl, whisk the cream cheese and salt until smooth. Fill a large bowl with ice and water.
3. Make the peach puree: Puree peaches in a food processor. Transfer ⅔ cup of the puree to a small bowl. Reserve the rest for another use. In a medium saucepan, combine sugar, lemon juice, and cardamom and bring to a boil over medium-high heat, stirring until sugar dissolves. Add to puree and let cool.
4. Combine the remaining milk, cream, sugar, and corn syrup in a 4-quart saucepan. Bring to a boil over medium-high heat and boil for 4 minutes. Remove from heat and gradually whisk in cornstarch slurry. Bring mixture back to a boil over medium-high heat and cook, stirring with a heatproof spatula, until slightly thickened, about 1 minute. Remove from heat.
5. Gradually whisk hot milk mixture into cream cheese until smooth. Add reserved ¾ cup buttermilk yogurt mix and peach puree. Pour mixture into a 1-gallon Ziploc freezer bag and submerge the sealed bag in the ice bath. Let stand, adding more ice as necessary, until cold, about 30 minutes.
6. Remove frozen canister in ice cream maker from freezer. Assemble ice cream machine and turn it on. Pour the buttermilk frozen yogurt base into the frozen canister and spin until thick and creamy.
7. Pack frozen yogurt into a storage container. Press a sheet of parchment directly against the surface and seal with an airtight lid. Freeze in the coldest part of your freezer until firm, at least 4 hours.