Adjust the oven rack to the middle position and heat the oven to 350 degrees. Pulse almonds, sugar, and salt together in a food processor until finely ground, about 1 minute. Add the egg and extracts and process until combined, about 10 seconds. Add the butter and process until no lumps remain, about 30 seconds. Spread the mixture evenly into the cooled tart shell. Sprinkle figs over almond mixture in even layer.
Remove the cooled pears from the poaching liquid and pat them dry thoroughly with paper towels.
Slice each halved poached pear crosswise into 1/4 inch thick slices from the narrow end. Do not separate the slices. Slide a thin spatula under the sliced pears and gently lay it in the center of the tart. Working with the remaining pear halves one at a time, slide the spatula under each pear and press on the slices to fan them towards the narrow end.
Lay the fanned pear halves attractively around the tart, with the narrow ends pointed towards the pear in the center.
Bake the tart on baking sheet until the almond filling is puffed, browned, and firm to the touch, about 40 minutes, rotating the baking sheet halfway through baking. Transfer the baking sheet to a wire rack.
Melt the jelly in a small saucepan over medium-high heat, stirring occasionally to smooth out any lumps. Brush the glaze over the pears and let the tart cool completely on the baking sheet, about 1 1/2 hours.
To serve, remove the outer metal ring of the tart pan, slide a thin metal spatula between the tart and the tart pan bottom, and carefully slide the tart onto a serving platter or cutting board.
Make Ahead: The tart dough, wrapped tightly in plastic wrap, can be refrigerated for up to 2 days or frozen for up to 1 month. If frozen, let the dough thaw completely on the counter before rolling it out. The poached pears, submerged in the poaching liquid, can be refrigerated in an airtight container for up to 3 days. The almond filling can be refrigerated in an airtight container for up to 3 days. Once baked, the tart can be wrapped loosely with plastic wrap and held at room temperature for up to 4 hours before serving.