In a large, flat bottomed pan, heat oil and butter over medium heat. Add onion and sugar,stirring occasionally for 20 - 30 minutes or until caramelized and golden brown. Stir in walnuts and cranberries. Remove from heat and set aside.
On a lightly floured surface, roll out pastry into 12" x 9" rectangle. Spread the mustard along the middle third of the pastry.
Spoon onion mixture over the mustard. Top with brie and pear slices.
Starting at the left top corner of the outside long edge of pastry, make diagonal cuts 1" apart, almost to the filling (like the left side of a V).
Repeat on the right side in the opposite direction. Alternating strips from side to side, fold over filling to resemble a braid, brushing each strip with beaten egg to secure. Brush top with beaten egg.
Transfer to parchment lined baking sheet. Bake in preheated oven for about 25 minutes, or until pastry is golden and pears are tender.
Remove baking sheet from oven and let cool slightly. Cut into 8 pieces.
May be made ahead up to 4 hours. Loosely cover with foil and set aside. Reheat in a 350F oven for 8-10 minutes and serve.