Preheat oven to 400º F.

Leaving roots intact, slice leeks in half lengthwise.

Rinse under cold water to dislodge any dirt between the layers.

Most dirt is where the leek changes from white to green, about halfway up. Cut off dark green parts and discard.

Slice the leeks into ¼ inch half-rounds all the way down to the root.

Discard roots.

Melt butter in a sauté pan over medium-low heat.

Add the leeks and cook until wilted and tender, about 15-20 minutes.

Remove from heat and season with ¼ teaspoon of salt.

Using a mandolin, slice skin-on potatoes into ⅛ inch thick rounds.

Place slices in a bowl of cold water, rinse for 10 seconds, and drain.

Place drained potatoes in a bowl and toss with olive oil, ½ teaspoon of salt, black pepper, and fresh thyme. Toss well and set aside.

Drain tomatoes in a colander. Working over the colander, use your hands to gently open the tomatoes and tear into large, flat chunks.

Place drained tomato pieces into a bowl. Toss with the remaining ½ teaspoon salt, fresh basil, ¾ cup Parmigiano-Reggiano, and heavy cream.

In an 8x8 inch casserole dish or a 9 inch cast-iron pan or a pie plate, spread ⅓ of the leeks in the bottom of the dish.

Layer with ⅓ of the potato slices, shingling them around in a circle until the surface is covered. Spread ⅓ of the tomato mixture over the potatoes. Repeat two more times.

Transfer casserole to the oven and bake 35 minutes.

Briefly remove from oven and sprinkle with remaining cheese.

Return casserole to oven and cook for an additional 25 minutes, until surface is bubbling and the cheese has browned.

Remove from oven and cool for 5 minutes and serve.