Roughly chop the carrot, celery and onion, then place them in a bullet blender with a generous drizzle of EVOO. Blend until creamy, adding a small splash of water if needed.
Heat a drizzle of EVOO in a medium pan, then add the soffritto. Cook over medium heat for 7 to 10 minutes, stirring regularly. Add a splash of water whenever it starts to dry out.
Pour in the passata, stir well and cook for 10 minutes. While the sauce cooks, drain the bocconcini, cut them into strips and set aside.
Season the sauce with salt and pepper to taste, then stir to combine.
Add the rice once the sauce is hot and start timing immediately. Cook over medium heat according to the packet instructions, around 16 to 18 minutes, stirring regularly.
Whenever the rice begins to dry out, add enough hot water to keep it moist without completely submerging it. During the final 2 minutes, increase the heat slightly and add another splash of hot water if needed. Taste the rice to make sure it is cooked.
Remove the pan from the heat. Tear the basil leaves by hand and stir them through the rice, then add at least 50 g of butter and Pecorino Romano to taste. Mix and toss vigorously until creamy and well combined.
Spread the rice evenly over a baking tray and allow the steam to escape. Chill in the freezer for 10 minutes or the refrigerator for 30 minutes, until completely cool and firm enough to shape.
While the rice cools, crack the eggs into a bowl and beat well. Prepare separate bowls of flour, beaten egg and breadcrumbs, then line a baking tray with baking paper.
Lightly wet your hands and take a portion of rice in your palm. Place a strip of bocconcini in the center, cover with a little more rice and shape into a ball or elongated oval. Coat in flour, dip into the egg, then cover completely with breadcrumbs. Place on the prepared tray and repeat with the remaining ingredients. If the supplì feel soft, refrigerate them for at least one hour before frying.
Heat the sunflower oil in a small, deep pot. Once the supplì are firm, lower them carefully into the hot oil using a strainer or slotted spoon. Fry in batches, turning gently with tongs, until crisp and golden brown. Transfer to a plate lined with kitchen paper and repeat with the remaining supplì.