Combine the basil and garlic in a food processor and mix while slowly adding the olive oil.
Add salt and pepper to taste. Continue this process until all the basil has been used. Reserve 1/4 cup of the pesto to mix into the rice.
Transfer the remaining pesto to a bowl and mix with the shredded cheeses. Cover with plastic wrap and set in refrigerator.
For the rice: In a medium saucepot over high heat, add the beef stock, butter and salt. When the stock reaches a boil, add the rice and stir it a bit to prevent sticking.
Let it come to a rolling boil and lower the heat a bit to medium. Stir the rice once in a while, you want the rice to absorb all of the liquid; it should take 15 to 20 minutes. Taste the rice; it should be slightly al dente but creamy.
Remove the pot from the heat, add the cheeses and add the 1/4 cup reserved pesto to the rice and stir thoroughly.
Pour the rice onto a baking sheet and spread into a thin layer. Cover with plastic wrap and let cool, preferably overnight in the fridge.
For the filling and assembly: Portion out the cooled rice into identical portions. We weigh ours to 1.6 ounces. Take one portion and form it into a semi-circle in the palm of your hand, making a cup-like shape to place the pesto and cherry tomato. Scoop about 1 tablespoons of the pesto-cheese mixture into the half ball and then 2 tomato halves. Take another portion of rice and flatten it into a disk shape.
Place that disk over the half ball to enclose the pesto and tomatoes in the center. Have a bowl of warm water nearby to dip your hands into; it'll make the process easier and less sticky.
Roll the ball around to enclose all the ingredients in the center and shape into a perfectly round ball. Repeat with the remaining ingredients.
Place the flour and 1 1/2 cups cold water into a bowl and add a pinch of salt.
Mix with a whisk until smooth and about the consistency of eggnog.
Place the breadcrumbs in a large bowl or baking dish.
Roll each ball in batter and shake off excess batter. Then roll each ball in the breadcrumbs. Repeat with all the balls.
Use a heavy duty pot with enough canola oil to submerge the balls in. Make sure the oil isn't near the top of the pot because it will overflow when you drop your balls in. Bring the oil to 350 degrees F, using a high temp thermometer if you have one. If not you can throw a small piece of white bread in to test; it should brown quickly and float when the oil is hot enough.
Use a slotted spoon or wire basket to lower your balls into the hot oil. Don't overcrowd the balls - maybe 2 or 3 at a time. Fry in batches until they are golden like arancini ("little oranges"), about 5 minutes.
Remove and place on paper towels to drain. Eat hot.
