Prepare the vegetables by mincing the garlic and onion, dicing the potatoes and celery, cutting the carrots evenly, and slicing the green beans into 1/2-inch pieces. Similar-sized potato pieces will soften more evenly than a mixture of tiny cubes and large chunks.

Combine the vegetables, broth, tomato paste, Italian seasoning, salt, and pepper in a 6-quart slow cooker, then tuck the chicken among the vegetables. The liquid should surround the ingredients, but it does not need to cover every part of the chicken.

Close the lid and select Low for 5 to 6 hours or High for 3 hours. At the end of cooking, the potatoes and carrots should yield to a fork, and the chicken should meet the thermometer target above.

Lift the lid away from your face, transfer the cooked chicken to a cutting board, pull it apart with two forks, and return the shreds to the pot. The chicken should separate readily into moist strands as aromatic steam rises from the stew.

Fold the chicken through the broth and vegetables, taste for seasoning, and add salt or pepper as desired before ladling into bowls. Each serving should contain tender vegetables and chicken with enough savory broth to spoon around them.