Heat oil in a 12-inch heavy skillet over moderate heat until hot but not smoking, then cook shallots and squash, stirring, until shallots are softened, about 5 minutes.
Add broth, brown sugar, sage, and salt, stirring until sugar is dissolved. Simmer, covered, stirring occasionally, until squash is tender, 8 to 10 minutes. Remove from heat and stir in vinegar, pepper, and salt to taste.
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