Warm a large skillet over medium heat, then add the olive oil.
Slice each chicken breast horizontally to create two thinner cutlets. Season both sides with dried oregano, half of the salt, and half of the pepper. Place them in the hot skillet and cook until they're nearly done (about 3-4 minutes per side), then take them out and set aside.
In the very same skillet, toss in the chopped bell peppers, jalapeño, and diced onion. Sprinkle with the rest of the salt and pepper.
Sauté these veggies until they start to soften (about 5-7 minutes). Then, stir in the minced garlic and cook for just another 20 seconds – be careful not to let the garlic brown!
Pour in the chicken broth and add the chopped tomatoes. Let it gently simmer uncovered for about 5 minutes, allowing the liquid to reduce a bit.
Stir in the fresh lime juice and chopped cilantro. Carefully return the chicken pieces to the skillet, spooning the flavorful pepper, onion, and tomato mixture over them.
Let it simmer for a few more minutes, just until the chicken is fully cooked through and reaches an internal temperature of 165°F (74°C).
Sprinkle the shredded mozzarella cheese evenly over each piece of chicken. Cover the skillet with a lid until the cheese is beautifully melted and bubbly (about 1-2 minutes).
Garnish with extra fresh cilantro and lime wedges, then serve immediately for a delicious, healthy meal!