Bring water and agar-agar powder to a boil over medium heat. Reduce heat and simmer until thickened, about 5 minutes. Remove from the heat and refrigerate until set, 20 to 30 minutes.
Cream vegan butter and coconut oil with an electric mixer on medium speed for about 2 minutes. Sift in powdered sugar, 1 tablespoon at a time, mixing after each addition to combine. Sift in flour and ground pistachios; stir to combine. Stir in vanilla and salt until incorporated, scraping the bowl as necessary. Stir in the prepared agar-agar; your dough will become wet but don't worry, it will harden in the refrigerator. Cover the bowl with plastic wrap and refrigerate, 8 hours to overnight.
Preheat the oven to 350 degrees F (175 degrees C). Line a work surface with parchment paper. Line a baking sheet with parchment paper.
Remove dough from the refrigerator and divide in half. Cover one half with plastic wrap and return to the refrigerator. Cover the remaining half with plastic wrap and roll on the prepared surface until it's 1/4-inch thick, working it into your hands to warm it up if it's too thick.
Cut out cookies with cookie cutter shape of choice. Rinsing off the cookie cutters with hot water between cuts will help remove any stuck-on cookie dough. Place cookies on the prepared baking sheet.
Bake in the preheated oven until the edges turn golden brown, 10 to 20 minutes, depending on the size. Cool on the baking sheet briefly before removing to a wire rack to cool, about 10 minutes.
Whisk aquafaba with an electric mixer until foamy. Whisk in powdered sugar, 1 tablespoon at a time, until fully incorporated and thick. Stir in rose water.
Dip cooled cookies into the icing and place back on the baking sheet. Sprinkle with chopped pistachios.