Preheat the oven to 400°F. Grease a 12-cup muffin pan. Melt the butter on low heat.
In a mixing bowl, stir together the buttermilk and cornmeal and let it sit for 5 minutes.
Add to the mixing bowl the rice, flour, sugar, salt, baking powder, and eggs, then stir until well combined. Pour in the melted butter and stir until well combined. The batter will be thick.
Slide the muffin pan into the oven for a minute to get it hot. Remove the pan, then evenly fill the muffin cups halfway up with the batter.
Bake uncovered for 20-25 minutes until just beginning to brown and an inserted knife comes out clean. Serve warm.
Notes
The original recipe called for whole milk and for the eggs to be separated, but I prefer buttermilk in my cornbread so thought it would be good here, too. And I didn't have time to separate the eggs, but the muffins were still tender and fluffy so it worked out just fine!