Preheat oven to 200 degrees F (93 C) and line two baking sheets with parchment paper. Place one in the oven (where you'll place cooked onion rings) and one on the counter (where you'll place dipped onion rings). Also set out a plate layered with a few paper towels.
In a large shallow bowl, mix together brown rice flour, cornmeal, cornstarch, salt, pepper, paprika, and cayenne pepper. Also add dairy-free milk to a separate small mixing bowl and set aside.
One at a time, dip the onion rings in the dry ingredients first, then the dairy-free milk, then back in the dry ingredients until well coated. Place on the parchment-lined baking sheet on the counter and repeat until all rings are dipped. Overlapping them a little is fine.
Heat a large cast iron or metal skillet over medium high heat and add enough oil to form a scant 1/2-inch layer. Heat for 3-5 minutes, or until a little flour sprinkled in bubbles and browns. At that point, add only as many onion rings will fit comfortably in the pan.
Fry on both sides for 1-2 minutes, turning down heat if browning too quickly. Flip with a fork and handle carefully as the coating can be fragile at this point.
Transfer cooked onion rings to the paper-towel lined plate to blot off extra oil, then onto the parchment-lined baking sheet in the oven to keep warm. Continue until all onion rings are cooked.