Divide the prepared pizza dough into 8 equal portions and roll each portion into a thin round, taking care not to tear the dough.
Heat 2 tablespoons olive oil in a cast-iron skillet over medium heat. Fry each dough round 1–2 minutes per side until puffed and cooked through, adding a little more oil if needed; transfer to a plate and repeat.
In a small bowl, whisk together 2/3 cup mayonnaise, 1/4 cup diced roasted red peppers and the minced garlic; stir in a little oil from the pepper jar if desired and set aside.
Preheat the grill to high (about 500–550°F / 260–288°C) for at least 10 minutes.
Season the chicken breasts all over with kosher salt, black pepper, 1 teaspoon smoked paprika, 1 teaspoon garlic powder and 1/2 teaspoon dried oregano.
Grill the chicken 5–6 minutes per side, or until an instant-read thermometer registers 165°F (74°C). Remove from the grill and let rest 5 minutes, then slice.
To assemble sandwiches, spread the red pepper mayo on one fried dough round, top with sliced grilled chicken, 2/3 cup grated Havarti, avocado slices, shredded lettuce and tomatoes; finish with another dough round if desired and serve immediately.