Preheat the oven to 220°C (425°F) with a rack in the lower third. Line a large baking sheet with parchment and dust with flour.
If the chicken is raw, season and cook it in a lightly oiled skillet over medium-high heat for 5 to 6 minutes per side, until it reaches 74°C (165°F). Rest and slice.
Mix the flour, Greek yogurt, baking powder and salt with a spoon, then knead on a floured counter for 1 to 2 minutes until smooth and only slightly tacky.
Roll or press the dough out thinly, about 5 mm (1/4 inch), on the tray.
Bake the crust alone for 8 minutes, until set and lightly golden.
Spread with the marinara, then add the chicken, sliced pepper, mushrooms and red onion. Top with mozzarella and oregano.
Bake for 10 to 12 minutes more, until the cheese is bubbling and the base is crisp underneath.