In a medium saucepan, combine the chopped strawberries, 1/2 cup sugar, and lemon juice. Cook over medium heat, stirring occasionally, until the strawberries break down and the mixture thickens (about 10 minutes). Let it cool to room temperature.

In another medium saucepan, whisk together the heavy cream, whole milk, and 3/4 cup sugar. Heat mixture over medium heat, constantly stirring until the sugar dissolves and the mixture warms up.

In a separate bowl, whisk the egg yolks until pale and smooth. Slowly pour some of the warm cream mixture into the egg yolks while whisking constantly to temper them.

Pour the tempered egg yolk mixture back into the saucepan with the remaining cream. Cook over medium heat, stirring constantly until the custard thickens enough to coat the back of a spoon (about 5-7 minutes).

Remove the saucepan from heat and whisk in the vanilla extract and softened cream cheese until smooth. Allow it to cool to room temperature, then refrigerate for at least 4 hours, or until completely chilled.

While the ice cream base chills, combine crushed graham crackers, melted butter, and cinnamon in a small bowl. Spread the mixture on a baking sheet and bake at 350°F (175°C) for about 8 minutes or until golden brown. Allow it to cool.

Once the ice cream base is chilled, churn it in an ice cream maker according to the manufacturer's instructions. During the last few minutes of churning, add the cooled strawberry sauce and graham cracker crust, folding them gently into the ice cream.

Transfer the ice cream to an airtight container and freeze for at least 4 hours, or until firm. Serve and enjoy your delicious Strawberry Cheesecake Ice Cream!