Preheat the oven to 350 degrees F. Line 18 muffin cups with paper liners. Melt the butter in a large skillet over medium heat. Add the onion, celery and 1/2 teaspoon salt and cook, stirring occasionally, until just softened, about 5 minutes. Stir in the chestnuts, sage and 2 teaspoons thyme. Spread out the mixture on a baking sheet to cool, about 20 minutes.
Meanwhile, whisk the flour, cornmeal, sugar, parsley, baking powder, poultry seasoning, baking soda and remaining 3/4 teaspoon salt in a large bowl. Whisk the milk, eggs and vegetable-chestnut mixture in a medium bowl; stir into the flour mixture until just combined.
Divide the batter among the muffin cups, filling them three-quarters of the way. Sprinkle with the remaining 1 teaspoon thyme. Bake until the tops spring back when gently pressed, 18 to 22 minutes. Let the muffins cool 5 minutes in the pans, then remove to a rack to cool completely.