Heat a saucepan over medium heat and add extra virgin olive oil to coat the bottom. Add the minced garlic and coarsely chopped onion and sauté about 2 minutes, until softened. Transfer the garlic and onion to the slow cooker.
Pat the 4 chicken thighs dry. Season both sides with the 1/2 teaspoon salt and 1/2 teaspoon black pepper.
In the same saucepan, add additional extra virgin olive oil to coat the pan and heat to medium-high. Brown the chicken thighs 3–5 minutes per side, until golden brown. Transfer the browned thighs to the slow cooker and discard the oil from the pan.
Add the remaining ingredients to the slow cooker: 1 cup roasted peppers (bottled or canned), the 15-ounce can of kidney beans (drained and rinsed), 1/4 cup pitted and sliced olives, the 2 large peeled and diced potatoes, the 2 medium peeled and diced carrots, 1 teaspoon dried tarragon, 1 cup chicken broth, and 1 cup tomato puree. Nestle the chicken into the mixture and stir gently to combine.
Cover and cook on low for 8 hours or on high for 4 hours, until the vegetables are tender and the chicken is cooked through.
Remove the chicken thighs, discard the skin (remove before serving), return the meat to the slow cooker or shred it into the stew, stir to combine, and serve.