Preparation

Make the dough:

• Combine the flour, sugar, and salt in a large bowl. Add half of the butter and gently toss with your hands to coat each piece with flour. Using a pastry cutter or two table knives, cut the butter into the flour until it’s the texture of coarse meal. Add the remaining butter, gently toss again to coat each piece, and quickly cut again until the larger pieces are about the size of large peas. In two or three additions, sprinkle about 1/3 cup ice water into the bowl, lightly tossing the mixture between your fingers to moisten it evenly. Stop adding water when the dough looks ragged and rough but holds together when you gently squeeze a small clump in your palm. Shape the dough into a brick (be careful not to knead it, just squeeze it gently into a solid mass). Cut the dough in half crosswise. Press each half into a flat rectangle about 1/2 inch thick and wrap tightly in plastic. Refrigerate the dough for at least 2 hours. (The dough can also be refrigerated for up to two days or frozen for up to one month. Thaw it overnight in the refrigerator before using.)

Make the topping:

• Heat a 12-inch skillet over medium-high heat for 1 minute and then add the olive oil and butter. Once the butter has melted, add the onions, thyme, and 1/2 tsp. salt and sauté until the onions are tender and lightly browned, 6 to 8 minutes. Taste the onions and add more salt if necessary—they should be well seasoned. Transfer the onions to a medium bowl and set aside to cool. (This may be done up to a day ahead; keep covered and refrigerate.)

Roll the dough:

• Remove one rectangle of dough from the fridge. Cut it in half crosswise. Working with one half at a time, roll it on a lightly floured surface into a 1/8-inch-thick narrow rectangle, about 14 inches long and 5 inches wide. (If the dough is too cold to roll easily, let it warm at room temperature on a baking sheet lined with parchment (two narrow rectangles should fit side by side), top with a second sheet of parchment, and refrigerate. Repeat with the remaining dough and refrigerate until you’re ready to prepare the tarts. (This may be done up to 4 hours ahead; keep covered. To conserve space, stack all four rectangles on one baking sheet between layers of parchment.)

Assemble the tarts:

• About 1 hour before serving, position oven racks in the upper and lower thirds of the oven and heat the oven to 375°F. About 40 minutes before serving, divide the onions into four equal portions and spread them in a thin, even layer on the dough rectangles, leaving a 1-inch border. (You may not need to use all the onions; a thick layer will weigh down the tart slices and they’ll flop when your guests pick them up.) Fold the exposed border over the onions—if the dough is too thick in the corners, trim as necessary. Brush the folded edges with the egg. Bake the tarts until the crust is well browned, about 30 minutes, switching the positions of the baking sheets halfway through. Remove the tarts from the oven and let cool briefly on the baking sheet. Cut each tart into eight slices and serve.