Heat the oil in an ovenproof 10-inch skillet over medium-high heat. Add the onion, carrots, and a pinch each of salt and pepper. Cook, stirring occasionally, until just tender, about 5 minutes.
Add the spice mix and cook, stirring, for 1 minute. Stir in the stewed tomatoes and bring to a boil. Simmer, stirring occasionally, until the juices release and evaporate, about 5 minutes. Stir in the chicken and cook until the meat just loses its pink color, about 3 minutes.
Unroll the crust and center over the chicken-vegetable filling in the skillet; poke 3 slits in the dough. Transfer to the oven and bake until the crust is golden brown, about 30 minutes.