To prepare broth, melt butter in a Dutch oven over medium heat. Add thyme and shallots; cook 10 minutes or until shallots are golden brown. Stir in chicken broth and porcini mushrooms; bring to a boil. Reduce heat; simmer, uncovered, 1 hour. Strain broth mixture through a sieve into a bowl; discard solids.
Return broth mixture to pan. Stir in shiitake mushrooms and sherry; cook 10 minutes over low heat. Stir in chives. Serve immediately.
Notes
Make the broth up to four days ahead and refrigerate. Just before serving, add the shiitakes, sherry, and chives, and heat the mixture.