How to Make It
In a large bowl, whisk 5 cups of cold water with the fine salt and sugar until dissolved. Squeeze and add 2 of the lemon quarters to the bowl. Add the chicken, cover with plastic wrap and refrigerate for at least 1 hour or up to 6 hours.
Meanwhile, add the olive oil to a 6-quart slow cooker along with the garlic, bay leaves, thyme and rosemary sprigs, black peppercorns, 1 1/2 teaspoons of kosher salt and the remaining 2 lemon quarters. Cover and cook on low until the oil registers 200° on an instant-read thermometer, about 1 hour. (The oil can be kept in the slow cooker on low for up to 6 hours.)
Remove the chicken from the brine and pat dry; transfer to a plate. Discard the brine. Using tongs, carefully lower the chicken into the warm oil. Cover and cook on low until the chicken registers 155° on an instant-read thermometer, about 40 minutes. Using a slotted spoon, transfer the chicken and garlic to a plate. Cover with foil and keep warm. Discard the lemon quarters and herbs. Using a ladle, transfer 1 cup of the infused oil to a measuring cup (do not use the peppercorns) and let cool until just warm, 15 minutes. Reserve the remaining oil for another use.
In a blender, puree the egg yolks with the lemon juice, 1 of the cooked garlic cloves and 1/2 teaspoon of kosher salt until smooth, about 30 seconds. With the machine on, gradually add the 1 cup of cooled oil until creamy, about 1 minute. Season with kosher salt.
Slice the chicken and serve with the aioli, bread and salad.