Whisk the sour cream with the chives, vinegar and a pinch of salt in a small bowl until smooth. Refrigerate until ready to serve.
Grate the potato, celery root and onion on the large holes of a box grater into a colander set over a bowl. Add 1 teaspoon salt and a few grinds of pepper, toss well and let stand 5 minutes. Squeeze handfuls of the mixture firmly with your hands to remove as much liquid as possible. Blot dry with paper towels, then transfer to a large bowl.
Add the beaten egg, matzo meal or flour, mushrooms, thyme and garlic to the potato mixture and toss well to combine. Scoop 1/4 cupfuls of the mixture and tightly pack into thin 3- to 4-inch pancakes. Transfer to a paper towel–lined baking sheet.
Preheat the oven to 250˚. Heat 1/4 inch vegetable oil or chicken fat in a large nonstick skillet over medium-high heat. Working in three batches, fry the latkes until deep golden brown, 3 to 5 minutes per side, reducing the heat as needed if the latkes are browning too quickly. Remove to a rack set on a rimmed baking sheet and sprinkle with salt and more thyme. Keep warm in the oven while you make the remaining latkes. Serve with the chive sour cream.