Melt the butter in a large skillet over medium heat. Add the onions, thyme, a large pinch of salt and a few grinds of pepper, and cook, stirring occasionally, 5 minutes; add the carrots, and continue to cook until the onions are softened and browned in spots, 3 to 5 minutes more. Add the wine, increase the heat to medium-high, bring the wine to a boil and cook until reduced by half, 2 to 4 minutes. Turn off the heat, stir in the half-and-half and mix in the chicken. Transfer the mixture to a large mixing bowl and refrigerate until cool, about 30 minutes.
Discard the thyme. Stir in the mayonnaise, parsley, snap peas and celery and season with salt and pepper. Divide the chicken mixture among the biscuits and serve immediately.