Tip:For balanced texture, be sure to slice the tomatoes, cucumbers, chicken, and bread about the same size.
Prepare a medium-high charcoal or gas grill fire. Clean and oil the grill grate.
Brush the bread on both sides with 2 Tbs. of the olive oil. Grill the bread until well marked, about 1 minute per side. Transfer to a cutting board, cut into 3/4-inch cubes, and set aside.
In a small bowl, mix the oregano, smoked paprika, 1/2 tsp. salt, and 1/2 tsp. pepper. Sprinkle evenly over the chicken breasts. Grill, turning once, until an instant-read thermometer inserted into the thickest part of each breast registers 165°F, 10 to 12 minutes total. Transfer to a cutting board, let rest for 5 minutes, and then cut into 3/4-inch cubes.
In a large bowl, whisk the remaining 6 Tbs. olive oil and the vinegar. Press the anchovy fillet and garlic clove through a garlic press into the bowl (or mince by hand). Add the bread, chicken, tomatoes, cucumber, red onion, and herbs, and toss well. Season to taste with salt and pepper and serve.