Step 1

Arrange a rack in middle of oven; preheat to 350°F. Heat 1 Tbsp. oil in a large skillet over medium-low. Add onion, garlic, and ½ tsp. Diamond Crystal or ¼ tsp. Morton kosher salt and cook, stirring often, until onion and garlic are softened and translucent but without taking on any color, 5–7 minutes. Remove from heat and stir in prepared horseradish, tomato paste, and red pepper flakes, and 1 Tbsp. thyme leaves. Transfer to a large bowl.

Step 2

Combine broth and remaining 2 tsp. Diamond Crystal or 1 tsp. Morton kosher salt in a small bowl or measuring cup, stirring to dissolve salt; set aside.

Step 3

Add potatoes to bowl with onion mixture and toss to combine. Arrange groups of 4–5 tomato slices and 4–5 potato and onion slices in alternating stacks in a 2–2½ qt. baking dish or medium cast-iron skillet so edges are aligned vertically, working your way around perimeter of dish before moving into center. The pieces should be very tightly packed; tuck leftover onion filling into any gaps.

Step 4

Pour reserved broth over vegetables and drizzle with 3 Tbsp. oil. Cover tightly with foil and bake until potatoes are tender and tomatoes are juicy, 40–50 minutes. Increase oven temperature to 425°, uncover baking dish, and continue to bake until liquid is reduced by about half, tomatoes are concentrated and jammy, and potato edges are golden brown and crisp, 30–45 minutes more. Let cool 15 minutes.

Step 5

Drizzle gratin with a little more oil if desired and sprinkle with sea salt and remaining 1 Tbsp. thyme leaves. Finely grate as much fresh horseradish over as you’d like.