Preheat your oven to 180°C / 350°F, suitable for all oven types. Prepare a 3 litre / 3 quart baking dish (about 22cm / 9" square and 6cm / 2.3" high) by ensuring it is clean and dry, ready to receive the potato mixture.
Peel the potatoes and cut them into cubes about 1 3/4 cm / 2/3" in size, aiming for even pieces so they cook at the same rate. Place the cubed potatoes into a large mixing bowl.
Add the milk, thickened cream, mayonnaise (or extra cream, if using the alternative), shredded cheese for mixing, thyme leaves, minced garlic, salt, and a generous grinding of black pepper to the bowl with the potatoes. Toss everything together thoroughly until the potatoes are evenly coated in the creamy, cheesy mixture and the seasonings are well distributed.
Tip the mixture into the prepared baking dish, spreading the potatoes out into an even layer so they cook uniformly. Cover the dish tightly with foil to trap the steam and help the potatoes become tender as they bake. Transfer the dish to the preheated oven and bake for about 60 minutes, or until the potatoes are soft when pierced with a knife or skewer.
Remove the foil carefully, taking care to avoid the hot steam. Sprinkle the shredded mozzarella cheese evenly over the top of the potatoes, creating a good, even coverage. Return the dish to the oven, uncovered, and bake for a further 25 minutes, or until the cheese on top is melted, bubbling, and a deep golden color.
Once baked, take the dish out of the oven and let it rest for 5 to 10 minutes so the creamy sauce can settle slightly and the dish firms up for easier serving. Garnish with a few extra thyme leaves, if you like, then spoon onto plates or serve straight from the baking dish at the table.