Add water and ice to a very large bowl. Place an open 1-gallon ziploc bag inside the bowl of ice water. Set aside.
In a medium saucepan, combine milk, heavy cream, dulce de leche, salt, spices, and vanilla extract. Whisk vigorously to combine the spices. Turn heat to medium and bring close to a boil, stirring mixture to help melt the dulce de leche. This will take anywhere from 7-10 minutes of constant stirring.
Once the dulce de leche has melted, reduce the heat to low and add the cornstarch slurry. Whisk the mixture until thickened, about 3-5 minutes.
Pour the mixture into the ziploc bag. Seal the bag and submerge the mixture into the bowl of ice water and allow it to come to room temperature, adding more ice as needed.
Remove bag from ice water and place in the freezer for at least one hour.
Remove bag from freezer, strain into an ice cream maker, and freeze according to manufacturer's instructions.