Pour the milk into a large saucepan. Add salt and 3 tablespoons of the sugar. Place over medium-high heat and bring to a boil. Stirring occasionally to make sure the milk does not burn along the sides or bottom of the pan.
Meanwhile, in a mixing bowl, whisk together remaining sugar and cornstarch. Add the eggs and whisk until smooth.
When the milk has boiled, slowly add a little bit of the milk into the egg mixture. Whisk constantly to distribute the heat. Continue to pour more milk until a couple tablespoons are left in the saucepan. Then pour the egg-milk mixture back into the saucepan.
Whisk constantly over medium heat until the custard has thickened. To cook the cornstarch out, heat the custard until it boils. Remove from heat. Whisk in butter one tablespoons at a time. Whisk until smooth. Add the vanilla paste and whisk until combined.
Pour pastry cream into a plastic wrapped sheet pan. Spread evenly. Cover with plastic wrap and freeze for 10-15 minutes until pastry cream has cooled to room temperature.
Remove from freezer and continue to cool in refrigerator until ready to use.