Remove and discard skin from rotisserie chicken. Remove meat from bones, reserving bones. Chop meat; cover and chill until ready to use.
Combine chicken bones, water, and next 5 ingredients (through thyme sprig) in a large Dutch oven; bring to a boil. Reduce heat, and simmer 1 hour. Strain stock through a sieve into a large bowl; discard solids.
Heat olive oil in pan over medium-high heat. Add green onions and garlic; sauté 1 minute. Add strained stock, and bring to a boil. Stir in rice; reduce heat, and simmer, uncovered, for 15 minutes or until rice is tender. Stir in reserved chopped chicken.
Combine lemon juice and eggs in a medium bowl, stirring well with a whisk. Gradually add 1 cup hot soup to egg mixture, stirring constantly with a whisk. Gradually add egg mixture to pan, stirring constantly. Cook over medium heat until slightly thick (about 2 minutes), stirring constantly. Remove from heat; stir in dill.