Add oil to a heavy bottomed large pot (I use a 3.5 quart dutch oven). Put over medium low heat, add a candy thermometer or deep fry thermometer to the edge and bring to 330 - 350 deg f. If the oil stops rising in temperature, turn the heat up a little.
In a large bowl, whisk together cornstarch, cassava flour, sugar, salt, and baking powder.
Add in softened butter, milk, eggs, and cheese. Mix this with your finger tips until a soft playdough-like dough forms. Add a tiny bit more milk if it is not coming together. Form into a ball.
Divide the dough ball into 4 equal pieces. Roll each of those pieces into balls.
Divide each of the four balls into 8 pieces. Roll each one into a ball. You will have 32 balls.
Fry 3-4 buñuelos at a time for 5-6 minutes until golden brown. Repeat for all of the dough balls.
Place on a plate lined with paper towels to drain excess oil. Serve when the balls have cooled.