Place the cottage cheese, frozen pineapple pieces, and frozen banana into a food processor or high-powered blender. Process until the mixture becomes uniformly smooth and creamy, pausing as necessary to scrape the sides and bottom of the container.
Add the honey, turbinado sugar, cold heavy cream, sweetened condensed milk, vanilla extract, and scraped vanilla bean to the blended fruit mixture. Continue blending for approximately 30 seconds, or until all of the ingredients are thoroughly incorporated and the mixture is smooth.
Transfer the finished mixture to a loaf pan and smooth it into the container. Cover the surface with plastic wrap to help minimize the formation of ice crystals. Place the pan in the freezer for approximately 4 hours, or until the dessert has become firm.
Check the consistency after the 4-hour freezing period. If the Dole whip is too firm to scoop comfortably, leave it at room temperature for a few minutes until it softens slightly. Scoop into serving bowls and finish each portion with an additional drizzle of honey.