Remove the meat from the chicken, discarding the skin and bones. Chop the chicken into bite-size pieces and set aside (you should have about 3 cups). In a medium saucepan, bring water and broth to boiling; reduce heat to low and cover to keep warm.
Meanwhile, in a 4- to 6-quart Dutch oven, cook shallots in hot oil over medium heat for 4 to 5 minutes or until just tender, stirring occasionally. Add rice. Cook and stir over medium heat 2 to 3 minutes or until rice begins to brown.
Slowly add 1 cup of the warm water mixture to the rice mixture, stirring constantly, until liquid is absorbed. Add another 1/2 cup of the liquid, stirring constantly until liquid is absorbed. Repeat two more times, using 1/2 cup liquid each time. Stir in asparagus. Add remaining water mixture, 1/2 cup at a time, stirring constantly and allowing the liquid to absorb before adding more liquid. (This should take 15 to 20 minutes total.)
Reduce heat to low. Add the shredded Parmesan cheese; stir until melted and smooth. Add chopped chicken; cook, uncovered, for 1 to 2 minutes or until heated through. Remove from the heat. Stir in arugula, tarragon and lemon zest. If you like, top with additional lemon zest and/or shaved Parmesan before serving.