Preheat the oven to 350°F. Lightly grease a baking dish or line a rimmed baking sheet with parchment paper.
Slice the tomatoes and fresh mozzarella into thin, similarly sized pieces.
Place the chicken breasts on a cutting board. Make four or five deep crosswise cuts in each breast, spacing them about ¾ inch apart. Be careful not to cut completely through the chicken.
Arrange the chicken in the prepared baking dish. Brush it thoroughly with olive oil and season with salt and black pepper.
Alternate slices of tomato and mozzarella in the cuts of each chicken breast.
Sprinkle half of the grated Parmesan over the chicken. Bake for 30–40 minutes, depending on the thickness of the breasts. The chicken is ready when its thickest part reaches an internal temperature of 165°F.
While the chicken is baking, cook the white basmati rice and wild rice blend according to their respective package directions. If their cooking times are different, prepare them separately and combine them after cooking.
Add the butter to the hot rice and season with salt. Stir gently, cover, and let it rest for 5 minutes.
Remove the chicken from the oven and sprinkle it with the remaining Parmesan. Let it rest for 5 minutes before serving with the buttery rice.