Place blackberries, sugar and water in a heavy small saucepan. Bring to a boil and allow mixture to cook and reduce by half. Mixture should be slightly syrupy. Remove from heat and cool slightly. Press mixture through a finemesh
strainer to remove most of the seeds. Discard seeds. Refrigerate syrup until cold.
Place half-and-half and heavy cream into a medium saucepan over medium heat. Bring mixture just to a simmer, stirring occasionally, and remove from heat.
In a medium mixing bowl whisk egg yolks until they lighten in color. Gradually add sugar and whisk to combine. Temper cream mixture into egg mixture until about a third of the cream mixture has been added. Pour in the remainder and return entire mixture to saucepan and place over low heat.
Continue to cook, stirring frequently, until mixture thickens slightly and coats the back of a spoon and reaches 170°. Pour mixture into a container and
place in an ice bath for 30 minutes. Stir in peaches. Place mixture into the refrigerator until temperature reaches 40° or below, about 4–6 hours.
Pour into an ice cream maker and process according to manufacturer’s directions. This should take approximately 25 to 35 minutes. Drizzle in blackberry syrup just to combine. Freeze for another 3 to 4 hours to allow the ice cream to harden.