To make the dough, whisk together flour, sugar, and salt in a large bowl. Cut in the butter with a pastry cutter or fingers until mixture resembles coarse crumbs. In a small bowl, whisk together one egg and milk. Add half of the egg and milk mixture to the dough and mix to incorporate. Mix in the remaining egg and milk, and make the dough into a flat disk. Wrap in plastic and refrigerate for at least 30 minutes. After refrigeration, roll out the dough into a 9 to 10-inch circle and place on a baking sheet lined with parchment paper. Brush the top surface of the dough with the apricot jam, and return it to the refrigerator until the pears are ready.
Preheat oven to 425°F. While the oven is heating, quarter and core the pears, then slice them lengthwise into quarter-inch slices. Place the pear slices in a fan shape on the chilled circle of dough, leaving a 1 1/2-inch border. Drizzle melted butter over the pears and then sprinkle with sugar and nutmeg. Gently fold the edge of the dough up and over the pears to form a rim. In a small bowl, beat the remaining egg and brush the rim and edges of the dough with the beaten egg. Place the galette in the oven and bake for about 25 minutes until the edges are browning. Let rest a few minutes before slicing.