In a medium bowl, mix the eggs, egg yolks, cornstarch, and 1/4 cup of the half-and-half. Set aside next to a sieve over a bowl.
Rub the vanilla paste into the sugar in a medium saucepan, then pour in the rest of the half-and-half, and salt.
Bring the sugar mixture to a full boil, remove from the heat then slowly drizzle about 1/2 cup into the egg mixture while stirring.
Stain the egg mixture, then pour it into the saucepan, and place the saucepan over medium heat.
Use a wooden spoon to stir until thickened, about 40 seconds, or until it reaches 180F. Don't rush this part, if the mixture does not reach the right temperature, it will not thicken. (might thin out completely)
Add the butter and stir until it's completely melted, then transfer to a bowl or a container.
Place a piece of plastic wrap over the creme and let cool at room temperature for one hour, then refrigerate for up to three days.