Melt the butter in a large skillet over medium heat. Add the onions and thyme. Cook, stirring occasionally, for 5 minutes.
Add the carrots, and continue to cook until the onions are softened, about 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 milk and mix in the chicken and peas.
Transfer the mixture to a large mixing bowl, and refrigerate until cool, about 30 minutes.
Later, stir in the mayonnaise, parsley and celery. Season with salt and pepper.
Divide the chicken mixture amount 4 pieces of bread. On remaining bread, spread on 1/4 tsp mayonnaise. Then, on the mayonnaise, place the spinach.
Top with remaining bread and grill on a panini press until both sides are golden brown.