In a 4 quart kettle, combine the 2 cups milk, heavy cream, sugar, corn syrup, and salt. Whisk to combine. Bring to a simmer over medium-low heat, 5-8 minutes.
While heating, whisk together the remaining 2 tbsp milk, egg yolks, and cornstarch.
When the milk and cream mixture begins to simmer throughout, remove from the heat and slowly whisk in the cornstarch slurry. Return to medium-low heat and whisk until simmering throughout and slightly thickened, about 1 minute.
Remove from the heat and add the vanilla bean paste. Whisk to combine and allow mixture to cool to room temperature. Cover and chill 6-8 hours.
Once chilled, pour into the freezer and follow the manufacturer's instructions. This usually takes 20-25 minutes. After churning, the ice cream can be immediately enjoyed as soft serve or poured into a container and frozen for an additional 4-6 hours for a harder consistency.
For the fudge sauce
In a medium bowl or glass measuring cup, combine the water, cornstarch, cocoa powder, and sugar. Whisk to combine. Heat in the microwave in 30-second to 1-minute increments until thickened, stirring between heatings, 2-3 minutes in total.
Once thickened, add the butter and vanilla. Whisk until smooth and enjoy immediately or store in the refrigerator for up to one week. Heat slightly before serving.