Preheat the oven to 400°F (200°C) and line a baking sheet with parchment paper or aluminum foil.
Slice the bread lengthwise into two halves and place both cut sides up on the prepared baking sheet.
Preheat a countertop griddle or skillet to medium and lightly spray or oil the surface.
Arrange the sliced red bell pepper on the griddle and cook until softened and slightly charred, then transfer to a small plate to cool.
Season the chicken breast with salt and pepper, then cook on the griddle over medium heat until cooked through; remove and let cool until safe to handle. For faster cooking, slice the breast in half lengthwise before cooking.
In a food processor, combine the grilled red pepper, 4 oz (half) of the Greek yogurt, and a dash of salt and pepper; process until smooth but with small flecks of pepper remaining.
Spread the red pepper yogurt sauce evenly over both bread halves.
Slice or shred the cooled chicken and arrange it over the sauce on the bread.
Arrange thin slices of tomato on top of the chicken, then sprinkle generously with shredded 6 Cheese Italian.
Bake in the preheated oven for 9 to 11 minutes, or until the cheese is melted and bubbly.
While the bread cools to room temperature, make the garlic yogurt aioli: combine the remaining 4 oz Greek yogurt, minced garlic, and a dash of salt and pepper in a small food processor and blend until smooth, about 10–12 seconds. Transfer to a small bowl.
Slice the baked bread, garnish with chopped parsley if using, and serve with the garlic yogurt aioli for dipping.