Add finely shredded carrots and whole milk to a blender. Process until completely smooth. 

Pour 1 cup heavy cream and vanilla into a large bowl, with a mesh strainer set on top. Set aside. 

Add contents of blender, remaining 1/2 cup heavy cream, sugar, cinnamon and salt to a small saucepan. Cook over medium heat, stirring to dissolve sugar. 

In a small bowl, whisk together egg yolks. 

When milk mixture begins to steam, remove 1 cup warm liquid and slowly drizzle into egg yolks while whisking continuously. Pour mixture back into small saucepan and continue to cook, stirring occasionally, until mixture coats the back of a wooden spoon or rubber spatula. 

Pour hot liquid through the mesh strainer into reserved heavy cream. Stir to combine. Cool over an ice bath to room temperature. Cover and chill thoroughly. 

Churn ice cream according to manufacturer's directions. 

Meanwhile, prepare frosting. In a medium bowl, beat together cream cheese and butter until smooth. Mix in vanilla and powdered sugar, followed by heavy cream. Transfer mixture to a piping bag or plastic storage bag with the end snipped off. 

Transfer ice cream to a freezer safe container, adding streaks of frosting throughout. Freeze until firm, 4-6 hours or overnight.