To start, season the chicken thighs with salt, pepper, and paprika, then heat half of the butter with the olive oil in a large skillet or dutch oven over medium-high heat. Add the chicken thighs, skin side down, and cook until golden brown, about 6 minutes per side. Remove the chicken from the skillet and set aside.
Add the remaining butter to the skillet and melt it over medium heat. Add the onion and cook until softened, about 5 minutes. Add the potatoes and carrots and cook for another 5 minutes. Sprinkle the flour over the vegetables and stir to combine. Cook for another 2-3 minutes, then add the thyme, chicken stock, and vermouth.
Stir to combine and bring the mixture to a simmer. Add the chicken thighs back to the skillet, skin side up, and cover with a lid. Cook for 25-30 minutes, or until the chicken is cooked through and the vegetables are tender.