Place the chicken thighs in the bottom of the slow cooker. Add the sliced leeks, carrot pieces, and minced garlic on top.
Pour the white wine and chicken broth over the ingredients. Sprinkle with thyme, salt, and pepper.
Cover and cook on low for 6 hours or until the chicken is tender and the carrots are soft.
In a small bowl, whisk the heavy cream with the cornstarch until smooth. Stir this mixture into the slow cooker.
Cover and cook on high for 20 more minutes until the broth thickens slightly.
Shred or leave the chicken in large pieces, then stir in the chopped parsley before serving.