Step 1Using a vegetable peeler, peel 2 wide strips of zest (3 to 4 inches long) from the grapefruit and place them in a small saucepan. Add half a cup granulated sugar and half a cup water to the pan and bring to a boil over high heat. Reduce the heat to maintain a bare simmer and cook until the zest is translucent and candied, about 10 minutes. Remove the pan from the heat and let zest cool in the syrup completely.
Step 2Heat the oven to 325 degrees. Line 2 large rimmed baking sheets with parchment paper.
Step 3Using a microplane zester, grate the zest from the rest of the grapefruit into a large bowl. Juice the grapefruit and pour half a cup juice into a glass measuring cup; discard any remaining. Add half a cup wine, the yogurt and vinegar and whisk to combine.
Step 4In a medium bowl, whisk together the flour and baking soda. In the bowl with the finely grated grapefruit zest, add 1 cup granulated sugar, the butter and salt and beat on medium speed of a mixer until light and fluffy, about 3 minutes (see Baker’s Note). Add half the dry ingredients and mix on low speed until almost combined. While mixing on low speed, pour in the liquid ingredients then add the rest of the dry ingredients and continue mixing until the batter is smooth.
Step 5Using a 1-ounce ice cream scoop or 2 tablespoons, scoop the dough into mounds on the prepared sheets, spacing them 2 inches apart.
Step 6Bake the cookies, rotating the baking sheets front to back and top to bottom halfway through, until lightly browned on the bottom, 20 to 22 minutes. Let the cookies cool on the baking sheets for 1 minute, then transfer to a rack and let cool completely.
Step 7While the cookies cool, drain the two strips of grapefruit zest from the syrup and transfer to a cutting board; reserve the syrup. Thinly slice the strips from short end to short end then transfer to a small bowl and mix with the remaining 2 tablespoons granulated sugar to coat.
Step 8In a medium, shallow bowl, whisk the powdered sugar with the remaining 6 tablespoons wine and 2 tablespoons reserved syrup until smooth. The glaze should be pourable but thick; add up to 2 more tablespoons wine to thin, if necessary.
Step 9Dip the top of each cookie in the glaze, letting the excess drip back into the bowl. Transfer the cookies to a rack, top with 1 or 2 strips of candied grapefruit zest while the glaze is still wet, and let stand until the glaze sets, about 10 minutes.