Combine the rice and water in a heavy 2 quart saucepan with a tight fitting lid. Bring to a boil, then cover and decrease the heat to maintain a simmer. Cook until the rice is tender and the water is mostly absorbed, about 25 minutes.
Drain off any remaining water and spread the rice on a sheet tray to cool. While the rice cools, prepare the honey mustard. In a small bowl, whisk together the honey and dijon mustard until well combined. Stir in the chopped gherkin and Sriracha. Cover and place in refrigerator until ready to use.
Once the rice is cool, place it in a bowl and add the beet, feta, oats, green onions, dill, eggs, chilies, salt, and pepper. With your hands, mix the ingredients until well combined. No need to be gentle. If the mixture does not hold together, add additional oats by the spoonful until it does.
Moisten your hands and, using a 1/3 cup measuring cup, form the mixture into 8 cakes about 2 1/2 inches in diameter. Place the formed cakes on a baking sheet, cover with plastic wrap and chill for 30 minutes.
When you are ready to cook the cakes, heat 1 tbsp of olive oil in a large skillet over medium heat until it shimmers. Cook the cakes 4 at a time, until golden brown, about 6 minutes. Gently flip the cakes halfway through cooking time. Repeat with the remaining 4 cakes.
Serve warm or at room temperature with the honey mustard sauce.