Heat Olive Oil (1/8 cup) in a large Dutch oven over medium-high heat. Add the Leek (2) and Mushroom (0.5 pound), and season with Kosher Salt (to taste) and Ground Black Pepper (to taste). Cook, stirring occasionally until the mushrooms are browned and crisped for 12 to 15 minutes.
Using a slotted spoon, transfer them to a medium bowl leaving any olive oil behind. They will have absorbed the oil as they cook, and released it back as they crisp. Add a bit more Olive Oil (1/8 cup) so there’s another 1/4 cup or so in the pot. Cook the remaining Leek (2) and Mushroom (0.5 pound), adding them to the bowl with the other mushrooms.
Without wiping the pot, add the Pearled Farro (1 3/4 cup) and season with Kosher Salt (to taste) and Ground Black Pepper (to taste). Cook over that same medium-high heat, stirring frequently until farro is toasted on the outside, it will go from pale golden brown to a toastier golden brown, about 5 minutes.
Add Vegetable Broth (4 cup) and 2 cups water. Season with Kosher Salt (to taste) and Ground Black Pepper (to taste).
Bring to a strong simmer and reduce heat to medium-low. Simmer gently, stirring occasionally, until farro is fully cooked and most of the liquid has been absorbed; 20 to 25 minutes. It should still look a bit loose, like risotto or a porridge. Remove from heat and add half the Fresh Chives (1/4 cup).
Toss Fresh Dill (1 cup), remaining Fresh Chives (1/4 cup), and the zest of a Lemon (1) together in a small bowl.
To serve, season farro with Kosher Salt (to taste) and Ground Black Pepper (to taste) and ladle into bowls. Top with Sour Cream (to taste), mushrooms and leeks, and dill mixture. Serve lemon wedges alongside for squeezing.