Preheat oven to 425 degrees with rack in upper third. Combine 4 teaspoons coarse salt and 1 teaspoon pepper in a small bowl.

Oil a 2 1/2-quart baking dish and sprinkle bottom with some salt-and-pepper mixture. Cut potatoes into 1/8-inch-thick slices with a handheld slicer or knife. Arrange slices vertically in baking dish; wedge in shallots and sprinkle thyme throughout. Brush with oil and season between potato slices with remaining salt-and-pepper mixture. Place baking dish on a rimmed baking sheet.

Whisk together eggs, broth, 1/4 teaspoon coarse salt, and 1/4 teaspoon pepper in a medium bowl. Add matzo meal and stir to combine. Pour mixture evenly over potatoes, tapping baking sheet on the counter to ensure mixture settles to bottom of baking dish. Cover with parchment, then with foil, and transfer to oven. Bake 45 minutes. Remove foil and parchment, rotate baking sheet, and continue baking until potatoes are tender when pierced with the tip of a paring knife, about 40 minutes more. Increase heat to broil and broil until golden brown, about 5 minutes. (Watch carefully to prevent overbrowning.) Garnish with thyme and sea salt. Let rest at least 15 minutes and up to 30 minutes.