Make the vanilla bean ice cream: In a medium saucepan, combine the cream, milk, sugar, and vanilla bean paste and cook over medium-high heat, stirring occasionally, until the sugar dissolves, about 5 minutes. Do not bring to a boil.
In a medium mixing bowl, whisk the egg yolks together, add the salt (if using), then temper the yolks by adding about 1/2 cup of the hot milk mixture at a time while continuously whisking the yolks. Repeat until all the hot milk has been added to the egg yolks, then pour the mixture back into the saucepan. Cook over medium heat, stirring continuously with a rubber spatula, until the mixture thickens and reaches 185°F, about 5 minutes.
Immediately strain the mixture through a fine-mesh sieve into a clean medium bowl. Using a large bowl, prepare an ice bath with about 3 cups of ice and enough water to cover the ice. Place the bowl with the ice cream mixture in the ice bath and stir once every few minutes until the mixture cools down to about 70°F. Be careful when stirring to avoid getting any water into the mixture. Remove the bowl from the ice bath and carefully wipe down the outside of the bowl to remove all water drops.
Freeze the mixture in an ice cream machine according to the manufacturer's instructions. Place the churned ice cream in the freezer to set up for at least 1 hour before serving. Store in an airtight container for up to 1 week.
Make the chocolate sauce: In a 4-quart saucepan, gently stir together the sugar, cocoa powder, espresso powder, salt, and evaporated milk to combine. While occasionally stirring, bring the mixture to a boil, then let the mixture boil, stirring constantly and making sure the chocolate mixture does not stick to the bottom and burn, until smooth and thickened, about 5 minutes.
Pull the mixture off the heat and gently whisk in the cubes of butter and vanilla extract until the butter is fully melted.
To serve, scoop the ice cream into dessert bowls and spoon the warm chocolate sauce over the ice cream.
Store the chocolate sauce in an airtight container in the refrigerator for up to 3 weeks. Reheat in a small saucepan over low heat, stirring frequently.