How to Make It
In a small bowl, beat 1 of the eggs. Cut the puff pastry in half crosswise. On a lightly floured work surface, roll out 1 piece of puff pastry to a 12-inch square. Cut a 1/2-inch strip from each edge of the pastry. Brush the edge of the pastry square with some of the beaten egg. Place the strips around the edge to make a raised border. Chill for 30 minutes. Repeat with the second piece of puff pastry.
Meanwhile, preheat oven to 450°. Using a large, sharp knife, cut off the ends of the squash until you reach the seeds. Scoop out and discard the seeds. Carefully slice the acorn squash crosswise into 1/4-inch thick rings. Arrange the rings on 2 rimmed baking sheets. Mix 2 teaspoons of the olive oil with the butter and honey and brush on the squash rings; season with salt and pepper. Bake for 15 minutes, or until the rings are browned on the bottom. Reduce the oven temperature to 375°.
In a medium bowl, mix the ricotta with the remaining egg and the heavy cream. Season with 3/4 teaspoon kosher salt and a generous pinch of pepper. In a medium skillet, heat the remaining 2 teaspoons of olive oil. Add the shallots and cook over moderately high heat, stirring occasionally, until golden. Stir in the sugar.
Brush the raised pastry borders with some of the beaten egg. Spread the ricotta filling in the pastry squares. Arrange the squash rings on top, browned sides up. Scatter the shallots over the tarts and bake for 30 minutes, or until the pastry is golden brown. Sprinkle the tarts with toasted pumpkin seeds and serve warm.