Preheat oven to 375º F.
Remove the stems from the mushroom caps, thus leaving an indentation in the underside of each cap to hold the filling later. Chop the stems and set aside.
Coat a large nonstick skillet with nonstick cooking spray and place over medium-high heat. When hot, add the whole mushroom caps and cook for about 1 minute on each side, until faintly browned. Place the mushrooms, stem end up on a baking sheet.
Return the pan to medium heat and add the broth. When it boils, add the mushroom stems and shallots and cook until most of the liquid has evaporated, 2 - 3 minutes. Scrape the mixture into a bowl and add the tomato, walnuts, mayonnaise, bread crumbs and tarragon. Stir to combine, then season with salt and pepper to taste.
Mound the walnut mixture among the mushroom caps, placing a generous tablespoon in each one.
Bake 15 to 18 minutes, until mushrooms are tender but hold their shape. If desired, place under a preheated broiler for a moment to brown the tops lightly. Serve warm or at room temperature.