Trim off fennel leaves; reserve. Cut off and peel stalks; slice into strips 1/4 inch wide. Trim out cores from bases, separate layers, then cut these into 1/4-inch-wide strips of convenient length.
Heat 1/4 cup oil in wide skillet; add fennel and garlic. Toss over medium heat 5 minutes. Add tomatoes, savory, fennel seeds, and salt. Cover; simmer 5 minutes. Uncover and cook until liquid has almost evaporated.
Add mushrooms, wine, and pepper; stir. Simmer 10 minutes, until liquid is nearly evaporated and mushrooms tender. Transfer to serving dish. Cool.
Add remaining tablespoon oil and season to taste. Sprinkle with basil and minced fennel leaves to taste.
Serves 4
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