In a saucepan, add sugar and 2 1/2 tablespoons of water. Cook over medium high heat, stirring constantly, until sugar dissolves. Stop stirring and cook until syrup is deep amber in color, swirling pan every so often to ensure even cooking. Remove from heat.
Carefully pour 3/4 cup heavy cream in to pan, being cautious, as cream will bubble. Stir until smooth. Return to medium heat, allowing to slightly bubble, about 5 minutes. Remove from heat and cover.
In a bowl, whisk together egg yolks, 2 tablespoons sugar, and remaining 1/4 cup heavy cream until smooth. Stir in milk. Pour in to warm caramel. Cook over medium heat, stirring constantly and keeping custard at a low simmer until custard is thick enough to coat the back of a spoon. Don't let it boil! Strain through a fine mesh sieve, stir in vanilla paste.
Set bowl over an ice bath to cool down quickly. Once mixture is cool enough, transfer to a container to store in the refrigerator until well chilled, at least 2 hours.
Pour custard in to ice cream maker and process according to manufacturer's instructions. Transfer to freezer safe container, freeze until firm, at least 3 hours.