Heat a large sauté pan, preferably one with a lid, over medium-high heat for one minute. Once the pan is very hot, add 1 tablespoon olive oil and 1 tablespoon butter, and heat them together for another full minute. Arrange chicken skin side-down and cook until deeply brown underneath, about 4 to 5 minutes. [Don’t crowd the chicken; depending on the size of your pan, you might need to do this in two batches.] While it brownes, season what is now the top side with additional salt and pepper. Once browned, flip the pieces over and brown on the second side, about another 3 to 4 minutes. Don’t skimp on the color, please. Transfer the chicken to a plate.
Prepare the buttered onions: Leave the pan on medium-high and add 2 tablespoons butter to fat and juices in it. Once melted, add the onions and season with 1 teaspoon kosher salt. Cook the onions for about 15 minutes total, or until golden throughout and darker brown at the edges, stirring every couple minutes. In the beginning, the onions are watery; once the water has cooked off and the onions begin to pick up color, I reduce the heat to medium for the remaining time. Carefully taste and season with more salt, if desired, and lots of freshly ground black pepper. Remove 1/3 cup of onions and set aside until the end.
Finish the chicken and rice: Add thyme and dried rice and cook with the onions for 1 minute. Add the wine, if using, and cook until it disappears, about 1 to 2 minutes. Scrape down the pan sides, pushing any dried rice and onions back to the bottom. Return the chicken thighs to the pan skin side-up, spacing them out. Carefully pour broth around chicken, the lower amount if you used wine, the full amount if you didn’t. Bring the pan to a simmer then reduce to the lowest simmer and cover. Cook rice and chicken together for 25 minutes, or until rice is tender.
To serve: Off heat, rest the dish for 5 to 10 minutes before serving. Scatter with reserved buttered onions and scoop it onto plates.
Do ahead: This keeps so well. I reheat it covered in an oven-safe dish at 350 degrees for 15 to 30 minutes, until the chicken is warm. To freeze, well, I’m just going to say it: I’ll pack it ingloriously in a freezer bag and press out the extra air. Defrost in the fridge for a day, if you have time, then rewarm as written above.