Bring the rice and water to the boil in a saucepan over high heat. Reduce heat to medium-low, cover and simmer until the rice is tender and the liquid has been absorbed, 20 to 25 minutes.
Preheat an oven to 180 C / Gas 4. Beat egg whites until foamy in a large glass or metal mixing bowl. Lift your beater or whisk straight up: the egg whites should form a sharp peak that holds its shape; set aside.
Melt the butter in a large saucepan over medium-low heat. Beat in the flour, salt and pepper; stir until the mixture becomes paste-like and light golden brown, about 5 minutes. Gradually whisk the milk into the flour mixture and bring to a simmer over medium heat. Cook and stir 5 minutes. Whisk the egg yolks in a bowl until broken, then whisk in the milk mixture a few tablespoons at a time until you have whisked in 1/3 of the milk mixture. Scrape the egg yolk mixture into the saucepan with the remaining milk and whisk until smooth. Cook 2 more minutes, stirring constantly.
Remove the milk sauce from the heat and stir in the rice, tuna, onion and lemon juice. Gently fold in the egg whites until evenly blended, then pour into a 1.5 litre casserole dish.
Bake in the preheated oven until slightly puffed, golden brown and has set in the centre, about 40 minutes.