Prepare an ice bath in a large bowl with water and ice. Stir 1/4 cup of milk and the cornstarch in a bowl to create a slurry and set aside.
2 cups whole milk, 4 teaspoons cornstarch
In medium saucepan over medium-high heat, whisk the remaining milk, cream, sugar, corn syrup, and salt together. Bring to a boil, then cook for 4 minutes.
1 1/4 cups heavy cream, 2/3 cup granulated sugar, 2 Tablespoons light corn syrup, 1/4 teaspoon kosher salt
Whisk in the cornstarch and milk slurry, then bring to a boil again, cooking and stirring for 2 minutes until thickened.
Place the cream cheese in a large bowl. Add 1/4 cup of the hot liquid and whisk until smooth. Whisk in remaining milk mixture. Stir in graham cracker crumbs and vanilla. Pour into plastic bag and submerge in ice water until chilled.
3 Tablespoons cream cheese, 1 cup finely crushed graham cracker crumbs, 2 teaspoons pure vanilla extract
Pour into ice cream maker and churn according to manufacturer's instructions.
Stir in graham cracker pieces, honeycomb candy, and mini chocolate chips, then pour into containers and freeze 4 hours to cure.
4 graham crackers, 2/3 cup crushed chocolate [honeycomb candy], 1/3 cup mini chocolate chips