In a large mixing bowl, combine the heavy cream, sweetened condensed milk, and vanilla extract. Use a hand mixer or a whisk to beat the mixture until it is well combined and slightly thickened, about 2-3 minutes.
Drizzle the chocolate syrup into the cream mixture and gently swirl it with a spatula to create a marbled effect. Be careful not to overmix; you want to see ribbons of chocolate.
Fold in the crushed cookies, ensuring they are evenly distributed throughout the mixture.
Pour the mixture into a freezer-safe container, smoothing the top with a spatula.
Cover the container with a lid or plastic wrap and freeze for at least 4 hours, or until the ice cream is firm.
Before serving, let the ice cream sit at room temperature for about 5-10 minutes to soften slightly for easier scooping.