Preheat the oven to 350 degrees F. Add the bagel cubes to a rimmed baking sheet and toss with 2 tablespoons of olive oil. Bake until the bread is toasted, about 15 minutes. Remove from the oven and let cool slightly.
Meanwhile, add the vinegar, mustard, garlic, shallot, the remaining 4 tablespoons olive oil and 1/2 teaspoon each salt and pepper to a large serving bowl and whisk to combine. Add the tomatoes, fennel and onion. Toss to coat everything in the dressing and let sit at room temperature for 15 minutes.
Once the bread is slightly cooled and the vegetables have marinated in the dressing, add the cucumber, basil and mozzarella pearls and reserved fennel fronds. Toss to combine.