In 2-quart saucepan, cook tomatoes in olive oil over medium-high heat 2 minutes. Stir in sugar and red wine vinegar; bring to a boil. Reduce heat; simmer 10 minutes, stirring occasionally (being careful not to break up tomatoes), or until thoroughly heated. Stir in salt; remove from heat.
Heat oven to 425°F. Grease bottoms of 24 regular-size muffin cups with butter. Unroll pie crusts. Using 2 1/2-inch round cookie cutter, cut 12 rounds from each pie crust, rerolling dough if necessary. Press 1 round in bottom and halfway up side of each cup. In small bowl, beat egg and water until well blended; brush on crusts.
Place 1 cube Brie cheese in each cup. With slotted spoon, place 2 tomatoes into each cup (reserve candied-tomato sauce). Top cups with half of the Parmigiano-Reggiano cheese; sprinkle with basil.
Bake 9 to 12 minutes or until crust is golden brown.
Top tartlets with remaining Parmigiano-Reggiano cheese. Serve with reserved candied-tomato sauce. Serve warm.