Mash bananas and brown sugar together in a bowl. Transfer to a small saucepan, place over medium heat, and cook until bananas begin to brown, about 5 minutes, stirring constantly. Gradually mix 1 cup milk into bananas and heat until mixture begins to steam. Stir honey and cinnamon into mixture until honey has dissolved.
Pour banana mixture into a blender no more than half full. Cover and hold lid down; pulse a few times before leaving on to blend. Puree until smooth. Blend remaining 1/2 cup milk into mixture until creamy.