Tie the stems of the cilantro and dill into bundles, then trim off the leaves, reserving ¼ cup of each for garnish.

In a large pot, combine both sets of stems, the chicken and the remaining ingredients. Bring to a boil, then reduce heat to medium-low and simmer for 45 minutes. Remove the garlic head and transfer the chicken to a plate and cool until easily handled.

Once cool, shred the chicken into bite-size pieces, discarding the skin, bones and cartilage. Set aside.

To make the soup, strain the broth into another pot or bowl, discarding the solids.

Wipe out the empty pot, then add the carrots, onion, butter and salt. Set over medium-high heat and cook, stirring occasionally, until the onion is browned, 10 to 12 minutes.

Add the vermouth, scraping up any browned bits, and cook until evaporated, 1 to 2 minutes.

Add the flour and cook, stirring constantly, for 1 minute.

Add 2 cups of the broth and stir until smooth, then add the remaining broth and bring to a simmer.

In a medium bowl, whisk the yolks. Continue whisking while slowly adding 1 cup of hot broth from the pot.

Whisk in the lemon juice, then return the mixture to the pot and whisk to combine.

Add the chicken and any accumulated juices and cook until just heated through (do not simmer). Taste and season with salt and pepper.

Serve with chopped cilantro and dill leaves.