Preheat oven to 400 degrees.
Line a cookie sheet with a reusable silicone mat or aluminum foil.
Spread 2 teaspoons (2/3 of 1 tablespoon) of the softened butter evenly on one side of the flour tortilla, making sure to reach to the edge. Place the remaining softened butter aside.
Sprinkle 2 teaspoons of sugar over the buttered tortilla surface, distributing evenly.
Move the tortilla to the lined cookie sheet and quickly flip onto the silicone mat. (The butter and sugar coating should be on the underside; the plain side is facing up.)
Peel, core, and slice the pears into about 1/4 inch wedges. Arrange the wedges on the tortilla, starting at the outer edge with the thicker part of the wedge facing out. Continue making slightly overlapped circles, moving toward the center of the tortilla with each round. End with a few pear pieces in the center of the tortilla so that the pattern resembles a rose.
Divide the remaining 4 teaspoons of butter into small pieces and place atop pears. Sprinkle the pears with the remaining 2 teaspoons of sugar.
Bake for 30 minutes or until the pear slices are a light golden color and the tortilla is brown and crisp. Don't panic if burned sugar/butter appears around the outside of the tart--that burned stuff should stay on the mat or foil when you remove the tart.
Remove the tart from the oven and allow to cool for 5 minutes. Lift the tart carefully with a spatula and place on a rack to cool. If any burned bits have adhered to the crust, gently remove them.
While the tart is warm, heat the apricot preserves in the microwave in a small bowl for 10 seconds to liquefy. Brush the preserves over the tart with a pastry brush.
Right before serving, sprinkle the tart with the pistachios. Cut into 4 wedges and serve.