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Preheat oven to 350 degrees.
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In a small bowl, whisk together sugar, lime juice, vinegar, and 3 tablespoons safflower oil; set vinaigrette aside.
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In a large bowl, toss bread cubes with olive oil until well coated. Spread bread cubes in an even layer on a baking sheet. Transfer to oven until golden and toasted, about 10 minutes.
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Transfer toasted bread cubes to a large bowl and add cucumber, tomatoes, ginger (if using), cilantro, mint, basil, jalapeno, and arugula. Drizzle vinaigrette over top, and season with salt and pepper; toss to combine.