Bring stock and the water to the boil in small saucepan. Reduce heat; cover and simmer.
Heat half of the oil in medium saucepan, add onion and garlic; cook, stirring until soft. Add rice; stir to coat in onion mixture. Stir in ½ cup simmering stock mixture; stir, over medium heat until liquid is absorbed. Continue adding stock, in ½-cup batches, stirring, until liquid is absorbed after each addition. Total cooking time should be about 20 minutes. Remove from heat; stir in cheeses. Cover; refrigerate 30 minutes.
Meanwhile, heat remaining oil in small frying pan, add beef; cook, stirring until browned. Stir in paste and the water; cook, stirring, about 5 minutes or until mixture thickens, season to taste. Cool.
Make tomato and balsamic sauce.
To make arancini, shape tablespoons of rice mixture into cup shapes in palm of hand; fill with 1 teaspoon beef mixture. Carefully roll rice into balls to enclose filling. Roll arancini in breadcrumbs.
Heat oil in deep medium saucepan; deep-fry arancini in batches until browned. Drain on absorbent paper. Serve arancini with tomato and balsamic sauce.
Tomato and balsamic sauce
Combine ingredients in small saucepan; bring to the boil. Reduce heat; simmer, uncovered, stirring occasionally, about 10 minutes or until sauce thickens. Season to taste.
You could use ½ cup leftover bolognese mixture to fill the arancini.