In the bowl of a stand mixer with dough hook attachment add the Ryze flour, the 1 to 1 gluten free flour, tapioca flour, finely ground corn meal, yeast, and salt. Mix until well combined.
Add the warm water, milk, honey, and oil and mix for 30 seconds. Add in the egg and mix on medium for 2 - 2 ½ minutes.
Add in just enough gluten free flour, in 1/8 cup increments, to make a mildly firm, but still pliable dough. Cover the bowl and let the dough rest for 20 minutes.
Flour a board or mat with gluten free flour. Knead the dough gently for 3 to 4 minutes. Add more flour to the board to keep from sticking. Flour the board before forming the dough into a rectangle, about 9" x 6.5".
Melt the butter and olive oil in a saucepan over low heat or in the microwave for 20 seconds. Pour into a flat bowl and let cool a few minutes.Cut the rectangle vertically into 1 ⅛ inch strips, 8 pieces. Set out a baking sheet or stone pan with an edge. Place close to your working area.
Carefully dip each strip of dough into the butter mixture to coat. Lay each butter coated breadstick on the pan, twisting as you set it down. **see note below. Continue to dip each strip in the butter mixture and set each on the pan, leaving a minimum of 1 ½ inches between each. Use an additional baking sheet if needed. Top with grated cheddar cheese pressed onto the tops. For the herb version, use crushed garlic brushed on top, or garlic powder sprinkled on top. Add dried basil and oregano. Then top with parmesan cheese. Press into the dough.
Cover with a towel and let rise on the pan in a warm place for 30-45 minutes. Preheat the oven to 425° F.
Bake in the lower third of the oven for one pan, for 18 -20 minutes or until puffy. For baking two pans, rotate midway to bake evenly. If needed, heat the oven to 450° F and move the baking sheet to just above the center rack and bake for 2-5 minutes more until golden brown and cheese is crisped.