Adjust oven rack to middle position and heat oven to 400 degrees. Line rimmed baking sheet with parchment paper. Whisk Boursin cheese in small bowl until smooth.
Dust counter lightly with flour. Unfold pastry and roll into 10-inch square. Cut pastry into four 5-inch squares. Space pastry squares evenly on prepared sheet. Divide Boursin evenly among pastry squares, mounding filling in center of each square, and spread into 3-inch circle. Brush edges of pastry with egg wash, then sprinkle with bagel seasoning.
Bake Danish until puffed and deep golden brown, 15 to 20 minutes, rotating sheet halfway through baking. Let Danish cool on wire rack for 15 minutes. Sprinkle with chives and serve warm or at room temperature. (Cooled Danish can be stored at room temperature for up to 4 days; if desired, rewarm in 300-degree oven for about 10 minutes.)
