Set out accompaniments, such as bread, cream cheese, cucumbers, radishes, beets, and cornichons to serve with the chicken. It's an ideal meal on a weekend afternoon.
Poach the chicken: In a medium pot, combine chicken, wine, thyme, 1 teaspoon salt, 1/4 teaspoon pepper, and enough water to cover by 1 inch. Bring to a simmer over medium-high, then reduce heat to a low simmer and cook chicken until cooked through, 18 to 20 minutes. Transfer chicken to a heatproof bowl. Strain poaching liquid through a fine-mesh sieve over chicken; discard solids. Let sit until cool enough to handle. Remove skin and bones and thinly slice chicken; discard liquid.
To serve: Whisk together buttermilk, yogurt, oil, and basil, then season with salt and pepper. Serve chicken with buttermilk dressing and other desired accompaniments.
Letting the chicken cool in its cooking liquid results in moist, tender meat.