Preheat oven to 400°F. Dry chicken well and season generously with salt and pepper on both sides. Heat 1 tablespoon butter in a large cast-iron or other ovenproof skillet set over medium-high. Add chicken, skin side down, in a single layer. To get an even sear, place another heavy skillet on top of chicken, pressing it down. Cook until skin is crisp and deep golden brown, 10-12 minutes. Remove chicken to a plate and set aside. It will not yet be fully cooked at this point.
Add mushrooms and leeks and cook, scraping up any brown bits from the pan and stirring occasionally, until leeks are wilted and mushrooms are light brown, about 7 minutes. Add garlic and thyme, season with salt and pepper and cook until garlic is fragrant, about 1 minute.
Stir in orzo and cook until toasty smelling, about 3 minutes. Pour in sherry and stir, until liquid is evaporated, about 1 minute. Add 3 cups broth, bring to a simmer and cook, stirring often, for 5 minutes.
Nestle chicken back in pan, skin side up and transfer skillet to oven; bake, uncovered, until chicken is cooked through (it should register 165ºF on a thermometer) and orzo is tender, about 15 minutes.
Remove from oven and transfer chicken to a platter. Stir orzo and, if dry, stir in the last 1/2 cup broth. Stir in lemon juice, zest and remaining 1 tablespoon butter. Fold in baby spinach, a handful at a time, until wilted. Taste orzo for seasoning. Serve with lemon wedges.