In a 3- to 4-quart pan over high heat, combine butter and shallots and stir until shallots are limp, about 1 minute. Add rice and stir until some of the grains are opaque, about 2 minutes longer.
Add lemon peel and 4 cups broth to pan and bring to a boil. Reduce heat and simmer, stirring often, until rice is tender to bite, about 15 minutes.
Meanwhile, in a 10- to 12-inch ovenproof frying pan over high heat, rub fatty side of chops in pan to lightly oil, then lay chops in pan and brown on all sides, about 5 minutes.
Put pan with chops in a 400° oven. Bake until meat is medium-rare (pink) in center of thickest part (cut to test), about 15 minutes, or medium (only slightly pink), about 20 minutes. Skim and discard fat from pan.
Return pan with chops to high heat. Add garlic, vinegar, and sugar. Shake pan and stir to release browned bits, then stir in 1/4 cup mint and remove from heat.
If rice is cooked before lamb is ready, turn heat to very low and stir in about 1/4 cup broth. Stir lemon juice and most of the remaining mint into rice, and if you want a creamier texture, a little more broth. Serve rice with lamb and pan juices, sprinkled with the final bits of mint.