In a large mixing bowl, use an electric mixer on medium speed to beat the softened butter, powdered sugar, and granulated sugar together for 3 to 4 minutes, until the mixture is very light, fluffy, and pale in color.
Beat in the rose water and fine sea salt until fully combined. Reduce the mixer speed to low and gradually add the 2 1/4 cups of all-purpose flour, mixing just until no dry streaks remain and a soft, crumbly dough begins to form.
Use a rubber spatula to fold in the 2 tablespoons of finely chopped dried rose petals until they are evenly distributed throughout the dough.
Turn the dough out onto a lightly floured surface and gently knead it 4 or 5 times, just until it comes together into a cohesive, smooth ball. Divide the ball in half and shape each half into a log about 1 1/2 inches in diameter. Wrap each log tightly in plastic wrap.
Chill the wrapped dough logs in the refrigerator for at least 2 hours, or until they are very firm. Preheat your oven to 325°F and line two baking sheets with parchment paper.
Unwrap one chilled dough log at a time. Using a sharp knife, slice the log into rounds about 1/4-inch thick, placing the rounds 1 inch apart on the prepared baking sheets. If desired, gently press a few whole dried rose petals onto the top of each cookie for garnish.
Bake the cookies for 18 to 22 minutes, rotating the pans halfway through, until the edges are just beginning to turn a very light golden brown. The centers should remain pale and look set.
Let the cookies cool on the baking sheets for 5 minutes before transferring them to a wire rack to cool completely. They will firm up as they cool into a classic, delicate shortbread texture.