1. In a blender, whirl raspberries and blackberries until smoothly puréed. Rub purée through a fine strainer into a bowl; discard seeds. Mix strawberries and 6 tablespoons sugar with purée. Set aside.
2. In a 2- to 3-quart pan, combine remaining 3/4 cup sugar and half-and-half. Stir over high heat until bubbles form at pan edge (scalding, about 180°), 5 to 8 minutes.
3. In a small bowl, whisk eggs to blend. Then whisk about 1/2 cup of the hot cream mixture into eggs. Return egg mixture to pan and stir over medium-low heat with a flexible spatula-scraper, scraping pan bottom and sides thoroughly for even cooking, until custard thickly coats a metal spoon (about 190°), 8 to 10 minutes.
4. Add vanilla to custard.
5. At once, nest pan in ice water and stir custard often until mixture is cold, 10 to 15 minutes.
6. Pour cold custard through a fine strainer into a bowl, then pour mixture into an ice cream maker (1-qt. or larger capacity), or strain directly into the maker; discard residue. Add berry mixture. Freeze according to manufacturer's directions until mixture is firm enough to scoop, dasher is hard to turn, or machine stops.
7. Serve softly frozen, or freeze airtight (see freezing facts).
Firming and Storing Ice Cream:
To get frozen desserts hard enough to scoop onto a cone, or to store them, transfer when frozen to an airtight container and put in the freezer at least 3 hours or up to 1 week.
If freezing with ice and salt, leave the frozen dessert in ice and salt up to 3 hours.
For best flavor and texture, serve frozen desserts within a week. On longer standing, icy crystals develop.
Nutritional analysis per 1/2 cup.