For the salad: Combine the white vinegar, sugar, and 1 teaspoon salt in a small saucepan with 1/2 cup water. Bring to a simmer over medium-high heat and then add the onions. Return the liquid to a simmer, then take off the heat and let stand 15 minutes; drain.
Put the arugula in a large salad bowl and add a few of the pickled onions and all of the cherry tomatoes; toss to combine. (Reserve the remaining pickled onions for another use.)
For the vinaigrette: Combine the olive oil, Champagne vinegar, honey, mustard, 1/2 teaspoon salt and a few grinds of pepper in a mason jar and shake vigorously until emulsified.
Add just enough vinaigrette to lightly coat the salad and toss well. Season to taste with salt and pepper. The remaining vinaigrette can be kept refrigerated up to 1 week.