Put sugar, evaporated milk, 1/2 cup of the whole milk, and eggs into a medium saucepan and whisk until well combined. Cook over medium-low heat, stirring constantly, until thickened, about 30 minutes. Strain the mixture through a fine-mesh sieve into a large cowl and set aside to let cool.
Once mixture has cooled, add pureed peach, peach soda, remaining whole milk, and a pinch of salt and stir well to combine.
Pour mixture into an ice cream maker and process according to the manufacturer's directions. Transfer ice cream to a sealable freezer-safe container and freeze.