Rinse rice until water runs clear. Drain.
Combine rice, water, and cinnamon stick in a saucepan. Bring to a simmer and cook 10 minutes.
Stir in milk and sugar. Simmer gently 15–20 minutes, stirring often so it doesn’t stick.
Remove cinnamon stick and stir in vanilla and salt.
Heat a large skillet over medium-high. Add butter and let it brown slightly.
Pour the rice mixture into the skillet and press down. Cook 4–6 minutes without stirring to brown the bottom.
Serve warm, scraping the browned bits into bowls.