Step 1Heat the oven to 350 degrees. Spread the almonds and pistachios on a baking sheet and toast them until golden, 10 minutes.
Step 2Lightly oil the sides of an 8-inch-square cake pan and line the bottom with an 8-inch-square piece of wafer paper. Set aside another 8-inch-square piece of wafer paper to be placed on top of the candy. (If wafer paper is unavailable, oil the bottom of the pan as well as the sides, and lightly oil an 8-inch-square sheet of wax paper to be placed on top of the candy; set aside.)
Step 3Whisk 2 cups of sugar and the water together in a small bowl. Pour the mixture into a 2-to 3-quart heavy-bottomed saucepan and stir with a spatula over medium heat to dissolve the sugar. When the sugar is completely dissolved, stop stirring and bring the mixture to a boil. Use a clean pastry brush dipped in water to wash down the sides of the pan to prevent crystallization of the sugar mixture.
Step 4While the sugar cooks, pour the honey into a 1-to 2-quart heavy saucepan. Place a candy thermometer in the honey (to get an accurate reading, you may have to tilt the pan slightly so that the honey covers the bottom of the thermometer) and cook over medium-low heat until the honey reaches 280 degrees and turns a darker color, about 16 minutes. It will look like a dark syrup. Remove it from the heat.
Step 5Meanwhile, whip the egg whites in a mixer fitted with a whisk attachment until frothy. With the mixer running, gradually add the remaining 3 tablespoons of sugar and continue to whip to stiff peaks, 1 to 2 minutes.
Step 6With the mixer on high, carefully pour the honey into the egg whites in a slow, steady stream. Turn off the mixer and check the temperature of the sugar (to get an accurate reading, you may have to carefully tilt the pan so that the sugar covers the bottom of the thermometer). After the sugar has reached 330 degrees, turn the mixer back to high and carefully pour the sugar into the egg white mixture in a slow, steady stream.
Step 7Continue to whip at high speed until the mixture thickens and becomes almost too stiff to whip, 5 to 10 minutes. The mixture will start out looking wet and glossy, but as it cools and stiffens the surface will take on a dull, matte appearance. Add the vanilla and whip about 1 minute longer, until the vanilla is well incorporated.
Step 8Working quickly, fold in the almonds and pistachios and scrape the nougat into the prepared pan. Smooth the top of the candy with a spatula, and top with wafer or wax paper (oiled side down). Use your hands or a clean spatula to press down and flatten the top. Let the candy stand 8 hours or overnight.
Step 9Run a knife around the edges of the pan to loosen the nougat. Remove the wax paper, if used, and turn the candy out onto a cutting board. (Cut through the edible paper.) Use a sharp, sturdy knife to trim off the edges and cut the nougat into 1-inch squares. Store the nougat pieces between sheets of wax paper in an air-tight containe