Melt 2 tablespoons butter in heavy large skillet over medium heat. Add shallots, tomatoes, garlic, basil, 2 teaspoons thyme and rosemary; saute 2 minutes. Add mushrooms; saute until very tender, about 15 minutes. Remove from heat. Season with salt and pepper. Cool. Preheat oven to 400°F. Butter baking sheet. Melt 3 tablespoons butter in small saucepan. Place kitchen towel on work surface. Top with 1 phyllo sheet (keep remaining phyllo sheets covered with plastic wrap and damp towel). Brush with melted butter. Top with phyllo sheet; brush with butter. Sprinkle with 1/2 teaspoon thyme. Repeat layering with 2 more phyllo sheets, brushing with butter and sprinkling with remaining 1/2 teaspoon thyme. Top with another phyllo sheet; brush with butter. Starting 1 1/2 inches in from 1 long side and 2 inches in from each short side, spoon mushroom mixture onto phyllo in 2 1/2-inch-diameter log parallel to long sides. Fold short sides over filling; brush folded edges with butter. Using towel as aid and starting on long side nearer filling, roll up strudel jelly-roll style. Using spatula, transfer strudel to prepared baking sheet, seam side down. Brush strudel with butter. Freeze 5 minutes. Bake strudel until golden brown, about 15 minutes. Cut strudel into slices. Serve with Arugula Pesto.