Peel and slice the carrots into 1/2-inch rounds. Cut the potatoes into 1-inch chunks. Dice the onion and mince the garlic. Trim any excess fat from the chicken and cut large chicken breasts in half so they cook evenly.
Place the carrots, potatoes, and onion in an even layer on the bottom of a 5–6 quart slow cooker. Lay the chicken pieces on top of the vegetables. Sprinkle the minced garlic, dried thyme, dried parsley, dried rosemary, salt, and black pepper evenly over the chicken and veggies.
Pour the low-sodium chicken broth over the contents of the slow cooker, then drizzle the olive oil evenly over the top. This helps keep the chicken moist and carries the flavors through the dish.
Cover and cook on LOW for 6–7 hours or on HIGH for 3–4 hours, until the chicken is cooked through (165°F) and the vegetables are very tender. Cooking on LOW is recommended for the most tender texture and best flavor.
When the chicken is done, carefully remove it to a cutting board. Shred it with two forks or slice it into bite-size chunks. Return the shredded or sliced chicken to the slow cooker and stir gently to combine with the vegetables and broth.
In a medium bowl, whisk together the Greek yogurt, cornstarch, and Dijon mustard until smooth. Ladle about 1/4–1/2 cup of the hot cooking liquid from the slow cooker into the bowl and whisk again to temper the yogurt so it doesn’t curdle.
Pour the tempered yogurt mixture into the slow cooker while stirring gently. Add the frozen peas and stir again to combine. Cover and cook on HIGH for 15–20 minutes, just until the sauce has thickened slightly and the peas are bright green and tender.
Turn off the slow cooker. Stir in the lemon juice and grated Parmesan cheese (if using). Taste and adjust seasoning, adding more salt, pepper, or an extra squeeze of lemon as needed to brighten the flavors.
Ladle the creamy chicken and vegetables into bowls. Serve as-is or over brown rice, quinoa, whole wheat noodles, or cauliflower rice for a lighter option. Garnish with fresh chopped parsley or chives if desired and serve warm.