Combine the basil and garlic in the bowl of a food processor and pulse to finely chop. Add the oil, vinegar, lemon juice, salt, and pepper, and process to blend. Transfer to a bowl and set aside while you prepare the salad.
Salad
Place the onion in a small bowl and cover with cold water. Let stand for 15 minutes. Drain and blot dry with a kitchen towel.
Prepare the grill for direct cooking over medium heat (350° to 450°F).
Prepare the grill for cooking over medium heat (400°F).
Lightly brush the corn with oil and lightly season with salt. Brush the cut-sides of the baguette with oil and lightly season with salt.
Brush the cooking grates clean. Grill the corn over direct medium heat, with the lid closed, until slightly charred and softened a bit, about 10 minutes, turning occasionally. During the last 3 to 5 minutes of cooking time, grill the baguette over direct medium heat, cut-sides down at first, until grill-marked and golden brown, turning once.
Brush the cooking grates clean. Grill the corn over medium heat, with the lid closed, until slightly charred and softened a bit, about 10 minutes, turning occasionally. During the last 3 to 5 minutes of cooking time, grill the baguette over medium heat, cut-sides down at first, until grill-marked and golden brown, turning once.
Transfer the corn and bread to a cutting board. When cool to the touch, lay the ear of corn flat on the cutting board, and, using a sharp knife, slice along the length to remove the kernels, rotating the ear after each cut until all the kernels are removed. Tear the bread into bite-size pieces.
Spread the arugula in a large shallow serving bowl. Scatter the tomatoes, corn and onion over the arugula, and top with the bread. Gently combine. Drizzle the vinaigrette over the salad and serve immediately.
