Heat a gas grill with all burners on medium. Brush the bread with 1/4 cup of the oil and season it with 1/4 tsp. kosher salt and a few grinds of pepper. Grill the bread on both sides, checking frequently, until nicely browned, 3 to 4 minutes per side. When the bread is cool enough to handle, cut it into 1/2-inch cubes.
In a small bowl, soak the shallot in the vinegar for 10 minutes. With a slotted spoon, transfer the shallot to a large bowl, reserving the vinegar. Sprinkle the garlic with 1/4 tsp. kosher salt and mash it to a paste on a cutting board with the side of a chef's knife. Whisk the mashed garlic, the remaining 1/4 cup olive oil, 1/4 tsp. kosher salt, and 1/8 tsp. pepper into the reserved vinegar.
Toss the bread cubes, tomatoes, cucumber, basil, mint, capers, and vinaigrette in the bowl with the shallot. Season the panzanella to taste with kosher salt and pepper and serve.