Wash the chicken thighs and pat them dry with paper towels. Season the chicken generously with salt and pepper. Heat the oil in a large heavy skillet over medium-high heat. Add as many of the chicken thighs, skin side down, as will fit without touching.
Cook until the skin is golden brown, about 10 minutes. Flip the chicken pieces and cook until the other side is golden brown, about 5 minutes more.
Remove the pan from the heat and spoon off all but about 2 tablespoons of the fat, leaving all the lovely brown bits behind. Return the pan to low heat, stir in the garlic and thyme and cook until the garlic starts to turn golden, about 1 to 2 minutes. Pour in the sherry, increase the heat to medium and bring to a boil. Let it boil a minute or 2, then return the chicken to the pan skin side up. Reduce the heat to low, pour in the chicken stock and bring to a simmer. Cover the pan tightly and cook 20 minutes. Flip the chicken pieces (or rotate them top to bottom if using a smaller pan) and cook until the chicken is very tender, about 20 minutes. Serve hot.
Note: If you like a mix of white and dark meat, substitute a 4-pound chicken cut into 10 pieces. Let the thighs, drumsticks, and wings cook in the sauce for 15 minutes before adding the browned breast pieces. If you'd like to use all white meat, cut the breasts in half crosswise, brown them as described above, but let the sauce cook down 10 to 15 minutes before adding the chicken breasts.
I like to use chicken thighs here-they won't dry out during cooking. If you like white meat, go ahead and use it, just cook it a little less. A paella pan, with its wide and shallow shape, comes in handy for this dish. If you have a smaller Dutch oven, brown the chicken in batches and rotate the pieces as they simmer in the sauce so the pieces cook evenly.