Bring first 8 ingredients to a boil in a large pot. Reduce heat, partially cover, and simmer 30 minutes. Drain broth, discarding solids; cool. Cover and chill.
Place fish on rack in poacher. Pour chilled broth into poacher. Add water, if necessary, to completely cover fish. Cover and cook over medium heat until liquid begins to simmer (about 25 minutes). Reduce heat until bubbles barely break surface. Cook 5 minutes or until thermometer inserted in thickest part of fish reaches 125°. Residual heat should continue cooking fish to a safe 140° temperature. (If unsure, insert a knife to see if flesh is opaque.) Remove fish from broth; peel away top skin. Cool to room temperature. Cover and chill. Serve with Basil Aïoli.