In a small saucepan, combine the grated carrot, half & half, cinnamon sticks (or ground cinnamon), nutmeg, ginger, and salt.
Place the saucepan over medium-low heat and bring the mixture to a very gentle simmer. You don't want the half & half to boil. Allow it to cook for about 10 minutes, stirring occasionally, so the carrot and spices can infuse into the half & half. The mixture will take on a slightly warm color and aroma from the carrots and spices.
Remove the saucepan from heat and allow the infused half & half to cool at room temperature for at least 20 minutes. You want the mixture to be cooled enough so it doesn't cook the condensed milk in the blender.
Once cooled, strain the mixture through a fine mesh strainer to remove the grated carrot and cinnamon sticks.
Pour the cooled infused half & half into a blender and add the sweetened condensed milk and vanilla extract.
Blend until the mixture is fully combined.
Transfer the creamer to a glass jar or airtight container and store it in the refrigerator until ready to use. Shake well before each use.