How to Make It
Preheat the oven to 425°. In a medium bowl, toss the flour with 1 teaspoon of the sugar and the salt. Cut in the cold butter until the mixture resembles fine crumbs.
In a small bowl, beat the egg with the milk. Sprinkle 2 tablespoons of the beaten egg over the flour mixture and stir. Working quickly, gather the dough into a smooth mass, squeezing it gently. Transfer the dough to a lightly floured surface and pat it into a 5-inch disk. Wrap in wax paper and refrigerate until firm but not hard, about 30 minutes.
On a lightly floured surface, roll out the dough to a 9 1/2-inch round, turning it clockwise an inch or two each time you roll to maintain an even shape. Transfer the dough to a heavy flat baking sheet. Fold up 1/4 inch of the edge of the dough to form a neat rim. Using a knife, score the rim in a decorative pattern. Brush the rim with a little of the remaining beaten egg. Brush 1 tablespoon of the apricot preserves over the bottom of the dough and refrigerate while you prepare the pears.
Peel, quarter and core the pears. Slice each quarter lengthwise into 5 thin wedges. Arrange all but 6 of the pear wedges on the dough in a spoke pattern, overlapping them slightly. Cut the remaining pear wedges and arrange them in the center of the tart to form a decorative rose. Brush the pear slices with the melted butter and sprinkle with the remaining 2 teaspoons of sugar.
Bake the galette in the middle of the oven for 25 to 30 minutes, or until the pears are tender and the pastry is crisp and golden. Carefully slide the galette onto a rack to cool slightly.
Stir the hot water into the remaining 1 tablespoon of apricot preserves and brush on the pears. Serve the galette at room temperature.