Cook Aromatics Heat 3 tablespoons of butter in a large pot over medium heat.
Add chopped leeks and carrots. Cook, stirring occasionally, until they begin to soften, about 5 minutes.
Add Broth and Seasonings Stir in the broth, dried dill, bay leaf, and season with salt and pepper. Add the potatoes. Cover and simmer for about 20 minutes, until the potatoes are tender but not falling apart. Discard bay leaf once done.
Sauté and Add Mushrooms In a separate pan, melt the remaining tablespoon of butter. Cook the mushrooms over medium heat until lightly browned, about 5 minutes. Add them to the pot.
Create the Cream Base Whisk together half-and-half and flour until fully blended. Slowly stir into the soup and let simmer a few more minutes until slightly thickened.
Finish and Serve Divide into bowls, top with chopped fresh dill or a small sprig, and serve warm. Enjoy!