Mix mochiko, sugar, and 1/3 cup plus 2 tablespoons cold water in a small bowl. Cover with foil. Bring 2″ of water to a simmer in a large pot. Fit with steamer basket; add bowl to basket. Cover pot with a tight-fitting lid. Steam until mixture is translucent and gummy, 20–22 minutes. Remove bowl and stir mixture until a thick, smooth, slightly sticky dough forms.
Meanwhile, schmear strawberries all over with bean paste, if using.