Spread carrots and potatoes over the bottom and up the sides of a 4-quart or larger slow cooker. Arrange chicken on top of the vegetables.
Bring broth, wine, garlic and salt to a simmer in a medium saucepan over medium-high heat. Pour over the chicken and vegetables. Cover and cook until the chicken is cooked through and vegetables are tender, 2 1/2 to 3 hours on high or 4 to 4 1/2 hours on low.
Add artichokes to the slow cooker, cover and cook on high for 5 minutes. Meanwhile, whisk egg, egg yolks and lemon juice in a medium bowl.
Transfer the chicken and vegetables to a serving bowl using a slotted spoon. Cover and keep warm.
Ladle about 1/2 cup of the cooking liquid into the egg mixture. Whisk until smooth. Whisk the egg mixture into the remaining cooking liquid in the slow cooker. Cover and cook, whisking 2 or 3 times, until slightly thickened and sauce reaches 160°F on an instant-read thermometer, 15 to 20 minutes.
Stir in dill and pepper. Pour the sauce over the chicken and vegetables and serve.