In a medium saucepan, mix 1 cup of the cream with the milk, 3/4 cup sugar, and a pinch of salt. Warm the cream mixture over medium-high heat, stirring occasionally, until the sugar dissolves and tiny bubbles begin to form around the edge of the pan, 3 to 4 minutes.
Stir in the black peppercorns. Cover, remove from the heat, and let sit for 1 hour. Taste and let sit longer if you want a stronger flavor.
Prepare an ice bath by filling a large bowl with several inches of ice water. Set a smaller metal bowl (one that holds at least 1-1/2 quarts) in the ice water. Pour the remaining cup of cream into the inner bowl (this helps the custard cool quicker when you pour it in later). Set a fine strainer on top. Whisk the egg yolks in a medium bowl.
Rewarm the cream mixture over medium-high heat until tiny bubbles begin to form around the edge of the pan, 1 to 2 minutes. In a steady stream, pour half of the warm cream mixture into the egg yolks, whisking constantly to prevent the eggs from curdling.
Pour the egg mixture back into the saucepan and cook over low heat, stirring constantly and scraping the bottom with a heatproof rubber spatula until the custard thickens slightly (it should be thick enough to coat the spatula and hold a line drawn through it with a finger), 4 to 8 minutes. An instant-read thermometer should read 175° to 180°F at this point. Don’t let the sauce overheat or boil, or it will curdle. Immediately strain the custard into the cold cream in the ice bath. Press firmly on the black peppercorns in the strainer with the spatula to extract as much flavor as possible.
Cool the custard to below 70°F by stirring it over the ice bath. Puree strawberries with the 1/2 cup sugar. Stir the strawberry puree, balsamic vinegar, and vanilla extract into the cooled custard.
Refrigerate the custard until completely chilled, at least 4 hours. Then freeze the custard in your ice cream maker according to the manufacturer’s instructions. Transfer to an air-tight container, and freeze for at least 4 hours or up to 2 weeks.