Broccoli / Carrots / Shallots - Prep as directed. Combine broccoli and carrots. Store shallots separately. (Can be done up to 5 days ahead)
Chicken - Slice in half. Cover with plastic wrap or parchment paper and then flatten with a meat tenderizer, until about 1/2” / 1.3 cm thick. Season with some salt and pepper and tenderize with a fork. (Can be done 1 day ahead)
Green onions - Slice green onions, keeping white and green parts separate. (Can be done 1 day ahead)
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Toss broccoli and carrots with cooking oil (portion for vegetables) and spread out onto a sheet pan. Season with some salt and pepper. Roast until tender, 25 to 30 minutes, shaking the pan halfway through cooking.
When vegetables are nearly tender, heat a skillet over medium-high heat. Add cooking oil (portion for chicken) and then chicken to heated oil. Sear chicken on both sides until golden brown and cooked through, 6 to 8 minutes total. Set chicken aside and return pan to heat.
Reduce heat under the pan to medium and add butter, shallots, and white parts of green onions, scraping up any browned bits on the bottom of the pan.
When butter is melted, add arrowroot powder and stir until no dry spots remain. Add vinegar and stock while whisking. Bring to a simmer and simmer until stock thickens slightly, ~2 minutes. Remove pan from heat. Stir in lemon juice. Season with some salt and pepper.
Serve chicken over roasted vegetables with pan sauce and green parts of green onions on top. Enjoy!