In a small bowl whisk the egg with 1/3 cup of the sugar until blended. Add the flour and whisk until thoroughly incorporated.

In a small heavy saucepan combine the butter, vanilla bean, orange zest, cardamom and pepper and cook over high heat until the butter is light golden brown and has a nutty aroma, about 3 minutes. Remove the vanilla bean, scraping the seeds from the inside into the melted butter. Discard the vanilla bean or reserve for another use. Allow the butter to cool slightly, then add to the flour mixture and whisk to combine. Allow to cool to room temperature, then refrigerate until thoroughly chilled, about 1 hour. This filling may be prepared up to 3 days in advance.

Halve the puff pastry sheet and roll each half out to approximately 1/8-inch thickness. Using a small plate as a guide, cut 3 (6-inch)circles out of each half of puff pastry with the tip of a sharp paring knife. Transfer the rounds onto 2 ungreased baking sheets, cover with plastic wrap and refrigerate for 30 minutes or up to overnight before proceeding.

Preheat the oven to 375 degrees F.

Using a sharp knife, cut approximately 1-inch off the tops of each pear, so that the remaining fruit is more or less spherical. Peel, halve, and core the pears. Rub them with the juice of half a lemon to keep them from discoloring. Place 1 pear half, cut side down, onto a work surface, and slice crosswise into about 1/8-inch thick slices. Do not separate the slices. Repeat with the remaining pear halves. Squeeze more lemon juice over the sliced pears. Reserve any uneven pieces and the end pieces separately.

Remove the puff pastry rounds from the refrigerator and place 1 heaping tablespoon of the butter filling into the center of each round. Using 1 pear half for each round, decoratively fan the slices in a tight, overlapping circle so that the pear slices cover the pastry round. The slices should not extend beyond the edge of the pastry. Any uneven slices or end pieces of pear may be cut into small wedge-shaped pieces and then placed in the center of the tartlets to form rosettes. Sprinkle 1/2 teaspoon of sugar over each tartlet and bake for about 30 minutes, or until the pears are tender and the tartlets are lightly browned around the edges.

While the tartlets are baking, heat the apricot preserves in a small saucepan (thin with a small amount of water, if necessary). When the preserves have melted, use a pastry brush to gently brush the top of each tartlet with some of the preserves.

Garnish with mint and serve warm or at room temperature with crème fraiche or vanilla ice cream, if desired.