Place the chicken in a dish with all but 1 tablespoon of the marinade and turn to coat. Let stand at room temperature for 15 minutes. Preheat the sandwich grill. Remove the chicken from the marinade and discard the marinade. Place the chicken on the grill, close the top plate, and cook until the chicken is opaque throughout, about 5 minutes. Transfer the chicken to a plate, keeping the grill on and wiping the grill plates clean.
Brush the crust sides of the focaccia pieces with 1 tablespoon oil. Brush the cut sides with the reserved one tablespoon of marinade. On 2 of the focaccia pieces, layer half the cheese, followed by the chicken, tomato slices, and basil leaves, dividing them evenly. Divide the remaining cheese on top.
Cover with the remaining 2 focaccia pieces, marinade sides down, and press gently. Place the panini in the grill, close the top plate, and cook until the bread is golden and toasted and the cheese is melted, 3-5 minutes.