Place non-dairy milk in a small saucepan on medium heat.
Whisk in flour and continue to whisk until the mixture turns thick and has a mayonnaise-like consistency.
Set aside to cool to room temperature until no heat is left in the mixture.
Place room-temperature shortening and margarine or butter in a stand mixer with brown sugar and mix on medium-high for about 3 minutes.
Add vanilla to the cooled milk/flour mixture and blend well.
Add milk mixture to butter/sugar mixture in a stand mixer.
Let frostiong mix on med/high for another 5 minutes until no sugar granules are detected.
Use promptly on cooled cake and refrigerate to store.
