Step 1: Finely chop 1 cup of the halved strawberries and set aside.
Step 2: In a medium bowl, sprinkle 1/4 cup sugar and 1 tablespoon fresh lemon juice over the remaining strawberries. Let stand for 30 minutes, and then process the strawberries in a food processor until completely smooth. If you prefer no seeds, pour through a fine-mesh sieve. Set aside.
Step 3: In a medium saucepan over medium-high heat, combine 2 cups whipping cream and 1 cup milk, and heat until scalded. Remove from the heat.
Step 4: In a bowl, using an electric mixer, beat 4 eggs until frothy. Slowly add 3/4 cup sugar and 2 teaspoons vanilla extract, and beat until thick and a pale lemon color.
Step 5: Gradually whisk 1 cup of the hot cream mixture into the egg mixture. Then gradually pour the egg-cream mixture into the saucepan while whisking constantly. Cook slowly over medium-low heat, stirring constantly, until the mixture thinly coats the back of a wooden spoon, 5 to 10 minutes. Do not overcook or the mixture will curdle.
Step 6: Remove from the heat, pour through a fine-mesh sieve into a medium bowl, and let cool.
Step 7: Add the pureed strawberries to the cooled custard and mix well.
Step 8: Pour the custard into an ice cream maker and freeze according to the manufacturer’s instructions just until the ice cream begins to thicken. With the machine running, add the finely chopped strawberries. At this point, the ice cream will be firm but not hard. You can either spoon it into a container and place in the freezer to harden or serve it immediately.