Combine the Champagne and white balsamic in a small bowl and place the red onions in the bowl to quickly "pickle" for 5 minutes or up to 10 minutes. The vinegar will turn a light shade of pink. Remove the onions and reserve. Whisk the mustard and salt into the vinegar. Slowly drizzle in the olive oil, whisking briskly and constantly to create a semi-emulsion.
Just before serving dress the baby arugula with just a few drizzles of the vinaigrette. Arrange the dressed arugula in layer in a shallow, wide bowl or platter. Add the pickled onions, sliced tomatoes, cheese and torn basil leaves over the top. Add a few more drizzles of the vinaigrette and fresh cracked pepper to taste. Serve immediately with crusty bread to mop up any dressing left on the platter/bowls.