Heat oil in skillet, add sliced leeks, and cook until leeks are wilted and starting to brown, about 10 minutes. Add carrot to the pan, and sauté 5 minutes. Stir in 1 cup broth, scraping pan to loosen any browned bits. Add leek mixture and remaining 3 cups broth, chopped green onions, dried herbs, salt, pepper, and thyme sprigs to slow cooker. Cover and cook on LOW for 2 hours.