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.
Note: Make the broth up to four days ahead and refrigerate. Just before serving, add the shiitakes, sherry, and chives, and heat the mixture.