Smoke is subjective. We enjoy smoked foods and very often we prefer to smoke them ourselves. This is because it's hard to achieve the balance where the flavor of the smoke accents and enhances an ingredient without taking over center stage. When I taste a smoked cheese I want to taste the cheese first and then have the flavor of smoke slowly uncurl on my palate, wrapping around the flavors of cheese, deepening them and making them even more alluring. Burrata is a great example of a fresh cheese that can be subtly enhanced by the flavor of smoke. It seems to make it even sweeter and creamier than it is on it's own, with that nice tang underlined by the richness of the smoke.
We are currently using BelGioioso burrata and gently cold smoke it for an hour using our tube smoker in a Weber grill. We put the cheese and whey in disposable lasagna pans set over pans of ice and set them inside the grill. We now smoke it for 30 minutes then flip them over and smoke them for 30 minutes more. We then store the cheese in the whey. The cheese, smoke and whey blend flavors and harmonize in the refrigerator.