Dutch oven or large pot
Read through the entire recipe before getting started. Preheat oven to 375ºF. Wash and dry all produce. Press garlic cloves with flat side of knife until smashed. Remove ends from shallot, peel and thinly slice into half moons. Cut potatoes in quarters, then slice into ¼" pieces. Remove ends from carrots, cut into HALF and in HALF again to form quarters. Cut lemon into ¼" slices. Tear any thick, woody stems from thyme, tender stems may remain. Remove chicken from package. Rinse and pat dry.
In a large Dutch oven or large pot, heat 2 Tbsp. oil until shimmering. Season chicken with salt and pepper. Add chicken, skin-side down and sear over high heat undisturbed until skin is golden brown and crispy, 3-4 min. Flip and repeat on other side for 3-4 min more. Transfer to a plate.
Add butter to pot and melt. Add garlic and shallot and cook over moderately-high heat, stirring occasionally, until tender and begin to brown 2-3 min. Add Dijon mustard and cook, stirring, until its oils begin to separate, 2-3 min. Add thyme and ¼ cup of water to deglaze pan and cook, scraping up any brown bits on the bottom.
Add potatoes, carrots and lemon to the pot and stir to coat with sauce. Return chicken to pot, nestling it in with the vegetables.
Cover and transfer to oven. Bake for 30-35 min, basting every 10 min until chicken is fully cooked and vegetables are tender. Remove from oven.
The USDA recommends cooking chicken to an internal temperature of 165°F.
Divide chicken between two plates. Serve with potatoes and carrots. Drizzle thyme-mustard jus from the pan on top. Enjoy!