Heat 1 Tbsp avocado oil in a large heavy-bottomed pot or Dutch oven over medium-high heat until shimmering.
Add the diced yellow onion and sauté, stirring occasionally, until softened, about 5 minutes.
Stir in the uncooked brown rice and minced garlic and cook, stirring, for about 3 minutes to toast the rice and bloom the garlic.
Pour in the chicken broth, then add the lemon zest, lemon juice, and chopped rosemary; stir to combine.
Carefully nestle the chicken thighs into the pot in a single layer so they sit in the liquid without overlapping.
Cover and bring to a full boil, then reduce heat to low and simmer, covered, for 40 minutes.
Remove the lid, add the broccoli florets, replace the lid, and cook for an additional 5 to 10 minutes until the broccoli is tender and the rice is cooked through.
Serve the chicken, rice, and broccoli immediately; optionally grate a little Parmesan over the rice before serving.