Separate mushroom stems from caps, coarsely chop stems, and slice caps about 1/4" thick. Make a stock: heat sunflower oil and 1 tablespoon of the butter in a large pot over low to medium heat. Add mushroom stems, onion, carrot, thyme sprig, 1 tsp salt, and 1/2 tsp pepper and sauté until soft, about 10 minutes. Add 6 cups water, bring to a boil, then reduce heat and simmer uncovered for 30 minutes. Strain, reserving the liquid.

Meanwhile, in another large pot, heat the remaining butter over low heat and add the leeks (or ramp stems, scallions or green garlic). Cook until they are about to brown, about 15 minutes. Add mushroom caps and cook until they also begin to brown, about 10 minutes. Add flour and cook, stirring constantly, for an additional minute. Add mead and stir, scraping bottom of pot, briefly, until well combined. Add the prepared stock, thyme, 1-1/2 tsp salt, 1 tsp pepper and bring to boil. Reduce heat, and simmer 15 minutes.

Remove from heat. Add milk (or half-and-half), and parsley (or ramp greens), season with salt and pepper to taste, and reheat if necessary (but not to a boil). If making ahead for a meal and presentation is a factor, add the parsley and/or ramp greens when soup is reheated. Otherwise, they taste fine as leftovers but don't look as bright green.