In a medium pot set over medium-high heat bring rice and 1 1/2 cups water to a boil. Reduce heat to a simmer and cook until water is absorbed and the rice is very soft. Set aside until warm and easy to handle.
In a medium sized bowl mix rice with flour, milk, egg, jalapeño, cilantro, baking powder, salt, and pepper.
Heat 2 inches of oil over medium-high heat until shimmering but not smoking.
Flour hands and form 2 tablespoon sized rice balls. Take a piece of cheese and push into the center, then roll again to ensure the cheese is completely covered with the rice mixture. Flatten slightly, dust with flour and set aside. Repeat with remaining rice mixture and cheese.
Fry balls for 1 to 2 minutes per side, until crisp and golden brown. Fry in batches for best results.
Transfer to a plate lined with paper towels to drain excess oil.
In a small bowl mash avocado and add juice of lime. Drizzle in avocado oil and season with salt and pepper. Mix until creamy and smooth. Serve fritters with tangy avocado sauce.