Heat olive oil in a skillet. Sauté ginger, sun-dried tomatoes, shallots, carrots, and celery over medium heat. Caramelize or cook until slightly browned. Slowly pour in balsamic vinegar while stirring. Reduce the mixture by half, then add the beef broth. Bring the sauce back to a quick boil. Season with salt and pepper to taste. Reduce heat to low and let the sauce simmer until thick enough to coat the back of a spoon. Vegetables should be tender.
Remove sauce from heat, and cool for 10 minutes. Place sauce into a blender and puree mixture. Pour the sauce back into the pot and cover—warm sauce right before serving.
Preheat grill for medium-high heat. Season steaks with salt and black pepper, and cook to desired doneness. I recommend medium-rare to medium (an internal temperature of 135-145 degrees F).
Clean the mushrooms with a damp cloth to remove debris from the surface. Thread four mushrooms onto each skewer lengthwise, from stem through the top. Brush with olive oil and season with salt and pepper.
Place mushrooms onto the grill and cook for 2-3 minutes, turn and cook for an additional 5 minutes, or until mushrooms are nice and tender.
Remove steaks from the grill and let them rest. Meanwhile, remove mushrooms from skewers and slice them into smaller pieces.
Top each steak with a little warmed sauce, the sliced mushrooms, and garnish with a rosemary sprig.