Place chicken breasts in a large bowl and add kosher salt. Fill the bowl with water to cover chicken breasts and stir to dissolve the salt. Allow the chicken to sit in the “brine” for at least 30 minutes.
Remove chicken from brine and pat dry with paper towels.
Slice a slit into the side of each chicken breast, being careful not to slice the chicken all the way through.
Divide the cream cheese, spinach, sun-dried tomatoes, and mozzarella between the chicken breasts and layer the ingredients inside (the order of layering isn’t important).
Use a toothpick to close up and hold each stuffed chicken breast together. Sprinkle both sides of the chicken with salt and pepper.
Preheat the oven to 400°F.
Heat olive oil in a large skillet over medium-high heat. Sear the chicken, browning for several minutes on each side. You may need to brown the chicken in batches so that there is space between the chicken while browning. Do not overcrowd. When the pieces are browned, remove them to a plate to fit the remaining pieces in the pan.
Return all of the chicken to the skillet. Sprinkle grape tomatoes over the chicken.
Bake for 10 to 15 minutes or until the chicken is cooked all the way through and registers 165°F.
Remove from the oven and allow chicken breasts to rest for about 5 minutes before serving.