Peel and slice the potatoes into 1/8-inch rounds, placing them in a large bowl of cold water to prevent browning—this is crucial for presentation. Dice the onion into 1/2-inch pieces and slice the kielbasa into 1/4-inch thick rounds. Crush the dried thyme between your palms to release its essential oils before measuring. Have all ingredients prepped and measured before you begin cooking, as the timing moves quickly once you start.

Heat the oil in a large skillet over medium-high heat. Once shimmering, add the kielbasa slices and cook for 3-4 minutes until they develop a nice golden color and release their fat. Add the diced onion to the pan and continue cooking for 4-5 minutes, stirring occasionally, until the onions become soft and translucent. The kielbasa's rendered fat will flavor the onions beautifully. Transfer the kielbasa and onions to a plate and set aside.

In the same skillet, melt the butter over medium heat. Once melted and foaming, sprinkle in the flour and whisk constantly for about 1.5 minutes to create a light golden roux—this cooks out the raw flour taste. Gradually whisk in the chicken broth, then add the heavy cream and milk in a steady stream while continuing to whisk to avoid lumps. Add the crushed thyme, salt, pepper, and nutmeg, stirring until the sauce thickens to a consistency that coats the back of a spoon (about 3-4 minutes). Remove from heat and stir in the sharp cheddar cheese until fully melted and smooth.

Drain the potatoes from their water bath and place them in a large pot of salted boiling water. Cook for 10-15 minutes until they're just fork-tender but still hold their shape—you want them partially cooked, not fully done, so they'll finish cooking in the oven without becoming mushy. Drain thoroughly in a colander.

Preheat your oven to 350°F. Pour a thin layer of the cheese sauce from Step 3 into a 9x13-inch baking dish. Arrange half of the parboiled potatoes from Step 4 in an overlapping layer over the sauce. Scatter half of the cooked kielbasa and onions from Step 2 over the potatoes, then sprinkle with half of any remaining cheese (reserve some for topping). Repeat with another layer of potatoes, the remaining kielbasa and onions, and most of the remaining cheese. Pour the remaining sauce over everything, then top with the final bit of cheese.

Cover the baking dish tightly with aluminum foil and bake at 350°F for 45 minutes. Remove the foil and bake for another 40 minutes until the top is golden brown and the edges are bubbling vigorously—I like to rotate the dish halfway through this uncovered baking time to ensure even browning. Let the casserole rest for 5 minutes before serving so the sauce sets slightly and becomes easier to scoop.