Preheat the oven to 350 degrees F. Butter an oval baking dish. Drain the leeks and rinse.
In a large skillet, melt the butter over medium heat. Add the leeks and thyme, season with salt and pepper and cook, stirring, until the leeks are softened and golden, 5 to 7 minutes.
Peel and slice the potatoes into very thin rounds. Sprinkle with salt and pepper. Layer half of the potatoes in the buttered dish, then layer half of the cooked leeks over the potatoes. Repeat the layers.
Return the skillet to medium heat and add the cream cheese, cream, chicken stock and garlic. Stir together, scraping up any brown bits from the bottom of the skillet. Sprinkle with the paprika and some salt and pepper. Bring to a boil, lower to a simmer and simmer for about 5 minutes so the cheese melts. Pour the cream mixture over the leeks and potatoes and top with the grated cheese.
Cover with foil and transfer to the oven. Bake for 40 minutes, then remove the foil and continue baking until the cheese is bubbly and golden, 15 to 20 minutes more. Allow to cool slightly before serving.