In a medium saucepan, combine the milk, cream and honey. Bring the mixture to a boil over medium-high heat. Meanwhile, whisk together the sugar and the egg yolks until the sugar is dissolved and the mixture is pale yellow, 30 seconds to 1 minute.
Once the milk is boiling, carefully stream the liquid into the egg mixture, whisking constantly to prevent curdling. Add the extracts and salt.
Chill the ice cream base over an ice bath, whisking constantly, 5 minutes, then transfer to a sealable container and refrigerate overnight. Make sure your ice cream maker base is in the freezer overnight while the base chills.
The next day, freeze the mixture in an ice cream machine according to manufacturer's instructions, then fold the rainbow cookies and raspberry jam into the ice cream to create swirls. Cover with plastic wrap and return to the freezer until set, 3 to 4 hours, then scoop and serve.