Step 1 Place the butter and flour in a mixing bowl. Using a pastry blender or your fingertips, cut or work the butter into the flour, leaving some of the butter in fairly large, irregular pieces. This will take 1 or 2 minutes. (Or mix for no more than a minute at medium-low speed in a stand mixer fitted with the paddle attachment.)
Step 2 Pour in 3 tablespoons cold water, stirring all the while with a fork until the dough begins to form clumps. (In the mixer, turn the speed to low and pour the water down the sides of the bowl, mixing for 30 seconds or less.) Add up to 1 additional tablespoon water if needed. Form the dough into a ball and wrap it in plastic. Flatten it into a disk and let rest, refrigerated, for 1 hour or longer.
Step 3Heat the oil or butter in a low-sided, heavy-bottomed pan. Add the onions and thyme. Cook over medium heat until the onions are soft and juicy, 20 to 30 minutes. Season with three-fourths teaspoon salt, or to taste, and cook for a few minutes more.
Step 4Transfer the onions into a bowl to cool. If the onions are very juicy, pour them into a strainer over a bowl to drain and remove the liquid.
Step 5Heat the oven to 375 degrees. Roll the tart dough out into a 14-inch circle on a lightly floured surface. Brush off the excess flour, transfer the dough to a large baking sheet lined with parchment paper, and put it in the refrigerator to firm up for 10 minutes or so.
Step 6Spread the cooled onions over the dough (removing the thyme sprigs as you go), leaving a 1 1/2 -inch border around the whole circumference of the dough. Fold the border up over the onions, overlapping the edges every 2 inches or so. Arrange the anchovies and olives evenly over the top. For a shiny, more finished appearance, mix together the egg and milk or water in a small bowl with a fork and brush the folded dough rim with it.
Step 7Bake until the crust is golden brown on the bottom, 45 to 50 minutes. Slide the tart off the pan onto a rack to cool. Serve warm or at room temperature.