Step 1

Broth: In a pot just large enough to hold hens snugly side-by-side, heat oil over medium. Add leeks, thyme, and a pinch of salt and cook, stirring occasionally, until leeks are soft but have not developed any color, about 5 minutes.

Step 2

Transfer mixture to a bowl. Season hens inside and out with salt and transfer to pot. Add enough water to just cover hens and bring to a boil, skimming any impurities from surface. Reduce heat to medium and simmer until hens are just cooked through, about 20 minutes.

Step 3

Spaetzle: Meanwhile, prepare an ice-water bath. Cook spinach in a pot of generously salted boiling water until wilted and darkened slightly, about 1 minute. With a slotted spoon, remove spinach and plunge into ice-water bath; let stand until cold, about 1 minute. Set pot of water aside for later.

Step 4

Thoroughly drain and squeeze spinach dry by pressing in a ricer or wringing out in a clean kitchen towel. Transfer spinach to a food processor with eggs, oil, nutmeg, cheese, and 2 tablespoons water; season with salt and pepper and purée until smooth. Add flour and pulse until no dry flour remains (dough should have the consistency of thick pancake batter).

Step 5

Broth: Transfer hens to a cutting board; strain broth through a fine-mesh sieve. Return broth to pot along with leek mixture and bring to a simmer over medium heat. Remove skin and bones from hens and shred meat into bite-size pieces.

Step 6

Spaetzle: Return pot of water to a boil. Press dough through a ricer or colander directly into water. Cook until spaetzle float to the surface, swell slightly, and are tender, about 5 minutes; drain.

Step 7

Stir 2 cups shredded meat and lemon juice into pot with broth. Spoon spaetzle into serving bowls and ladle broth mixture over top. Garnish with dill, a drizzle of oil, and serve with lemon wedges.