Set a large deep skillet with a lid over medium-high heat so it is hot when the chicken is ready.

Put the chicken pieces in a medium bowl and sprinkle with 3/4 teaspoon of the salt. Toss to coat the chicken evenly.

Add the olive oil to the hot skillet and swirl it across the bottom.

Add the chicken to the skillet in a mostly even layer. Cook without stirring for 3 to 4 minutes, until the bottom side has browned in spots.

Stir the chicken and cook for 2 minutes more, until the pieces look mostly opaque on the outside but are not fully cooked through. Transfer the chicken and any juices to a plate.

Reduce the heat under the skillet to medium. Add the chopped onion to the same skillet and cook for 3 minutes, stirring often, until it starts to soften.

Add the zucchini and the remaining 1/2 teaspoon salt to the skillet. Cook for 4 minutes, stirring every minute or so, until the zucchini is just starting to soften and pick up a little color.

Add the rinsed rice and dried oregano to the skillet. Stir for 1 minute, until the rice looks glossy and is mixed well with the vegetables.

Pour the chicken broth into the skillet and add the lemon zest. Stir once, scraping the bottom of the skillet so any browned bits loosen into the liquid.

Return the chicken and any juices from the plate to the skillet. Spread the chicken pieces across the rice mixture so they cook evenly.

Bring the liquid in the skillet to a gentle boil. As soon as it boils, cover the skillet with the lid and reduce the heat to low.

Cook, covered, for 18 to 20 minutes, until the rice is tender and most of the liquid has been absorbed. If you lift the lid, the rice should look moist but not soupy.