Place the chicken in a large pot (see Note 1) with the garlic, 2 teaspoons kosher salt, and 10 cups of water. Add more water if needed to ensure the chicken is covered by 1 inch.
Bring to a boil. Reduce the heat and simmer, uncovered, for 30 to 40 minutes, or until the chicken is cooked through and can be shredded easily. Periodically skim any foam from the top of the water.
Carefully remove the garlic and transfer to a small bowl to cool. Remove the chicken from the pot and transfer to a bowl. Once the chicken is cool enough to handle, pull the meat from the bones and shred it. Discard skin and bones or reserve for stock, if desired.
Once the garlic is cool enough to handle, carefully pop the cloves from the paper and mash them into a paste.
Skim off any remaining foam or scum from the pot. Replenish the pot with enough water to cook the pasta and add 1 teaspoon kosher salt. Return to a boil. See Note 2.