Preheat the oven to 350°F (175°C).

Spread the chickpea flour on a rimmed baking sheet.

Toast the chickpea flour for about 15 minutes, stirring occasionally, until lightly browned.

Reduce the oven temperature to 325°F (160°C).

Sift the confectioners sugar and all-purpose flour into the bowl of a stand mixer.

Carefully sift in the hot toasted chickpea flour.

In a saucepan, bring the butter just to a boil with the salt.

Turn the mixer to its next-to-lowest speed.

Slowly drizzle the hot butter into the flour mixture.

Continue beating the dough at this speed until it forms a solid ball, about 20 minutes.

Scrape the dough out onto a sheet of parchment paper.

Cover with another sheet of parchment paper.

Roll out the dough into a 3/4-inch-thick rectangle.

Peel off the top layer of parchment paper.

Transfer the dough to a baking sheet.

Refrigerate until firm, about 20 minutes.

Line another baking sheet with parchment paper.

Cut the dough into 1-inch squares.

Arrange the squares at least 1 inch apart on the lined baking sheets.

Bake for about 10 minutes, until the squares are lightly browned at the edges.

Rotate the baking sheets from front to back and top to bottom halfway through baking.

Transfer the shortbreads to a rack.

Let cool to room temperature before serving.