Line 2 baking sheets with parchment paper.

Beat butter, 1/2 cup sugar and vanilla in a large bowl with an electric mixer on medium speed, scraping the sides of the bowl as needed, until combined. Add rye; beat until combined.

Whisk all-purpose flour, pastry flour, cardamom, baking powder and salt in a medium bowl. Add the flour mixture, 1/2 cup at a time, to the butter mixture, mixing until just combined. Shape the dough into a log about 1¾ inches wide and 12 inches long. Wrap in parchment paper and refrigerate until firm, about 1 hour.

Meanwhile, using a sharp knife, cut peel off grapefruit, leaving some of the white pith still attached. (Reserve the grapefruit flesh for another use, such as fruit salad.) Place the peels in a small saucepan and add 2 cups water. Cover and bring to a boil over high heat. Drain the water from the pan. Repeat two more times.

Add 1/2 cup sugar and the remaining 1/4 cup water to the drained peels and bring to a boil over high heat. Reduce heat to a simmer and cook until the peels are deep pink or orange and slightly translucent, about 20 minutes. Remove the peels from the pan with a slotted spoon, allowing the excess liquid to drip off; transfer to a wire rack to dry, about 1 hour.

Meanwhile, preheat oven to 350°F.

Slice the cookie dough into 1/4-inch-thick rounds and place the rounds 2 inches apart on the prepared baking sheets. Use a fork to press a crosshatch pattern into the tops of the cookies. Bake until the edges are lightly golden brown, 12 to 15 minutes. Transfer the cookies to a wire rack to cool completely, about 20 minutes.

Cut the candied peels into thin strips or dice. Toss with the remaining 1/2 cup sugar. Remove the peels from the sugar.

Place chocolate in a microwave-safe medium bowl. Microwave on Medium for 1 minute. Stir, then continue microwaving on Medium, stirring every 20 seconds, until melted.

Dip half of each cookie in the melted chocolate and return to the wire rack. Top the chocolate-dipped part of each cookie with some of the candied grapefruit peel. Let the chocolate set, about 1 hour.