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To make the Sugar Plum Cookies: Preheat the oven to 350°F with racks in the upper and lower thirds. Line two baking sheets with parchment, or lightly grease.
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In the bowl of a stand mixer fitted with the flat beater attachment or working in a large bowl with an electric hand mixer, beat the butter, sugar, vanilla, and salt until smooth, 2 to 3 minutes at medium speed.
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Add the flour and continue mixing on low speed until the dough comes together and no dry spots remain, about 30 seconds. Use a flexible spatula to scrape the bottom and sides of the bowl to ensure everything is thoroughly combined.
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Scoop the dough into tablespoon-sized mounds (about 18g each); a Tablespoon Scoop works well here. Arrange the mounds evenly on both baking sheets. (You can place the mounds fairly close together; they won't spread much.)
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Bake the cookies for 14 to 18 minutes, rotating the baking sheets from top to bottom and front to back halfway through, until the cookies feel set when lightly touched on top and are lightly browned around the edges. While the cookies bake, prepare the simple syrup.
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To make the simple syrup: Place the water and sugar in a small, microwave-safe bowl. Heat in the microwave, in 30-second increments, stirring between each, until the sugar is fully dissolved. (Alternatively, this can be done in a small saucepan on the stovetop over medium heat.)
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Reserve 1 tablespoon (14g) of simple syrup and place it in a small bowl (this will be used to candy the stems), then stir in the vanilla and ube extract into the remaining liquid. Set both bowls aside to cool to room temperature.
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When the cookies are done baking, remove from the oven and allow them to cool completely on the baking sheets, about 45 minutes. While the cookies cool, prepare the candied stems.
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To make the stems: Dip individual mint leaves or small pieces of rosemary sprigs into the reserved 1 tablespoon of clear simple syrup, shake to remove any excess liquid, then sprinkle with granulated sugar. Set aside to dry on a cooling rack, about 1 1/2 hours, while you shape and assemble the cookies.
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To shape the cookies: When the cookies are completely cool, cradle one cookie, flat side-up, in the palm of your hand and use a melon baller, rounded teaspoon measuring spoon, or a small knife to gently scrape a shallow 1" wide hole in the center. Work slowly and start at the surface before carefully carving a deeper hole. (This hole will be filled with jam during assembly.) Save the scraped-out cookie crumbs; you’ll use some for the filling.
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To make the sugar plum filling: In a small bowl, place the plum jam. If it contains large pieces of fruit, use scissors or a paring knife to break up the plums into pieces that are 1/2" or smaller. Add the cookie crumbs and stir to combine. (The cookie crumbs help thicken the jam, which will make assembly easier.) Transfer the filling to a piping bag or a zip-top bag fitted with a 1/2" round piping tip.
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To assemble the Sugar Plum Cookies: Place all the ingredients for the sugar plum coating in a small bowl and stir to combine. Set aside.
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Fill the carved-out hole of each cookie with filling, then sandwich the two halves together. Only pipe enough filling to fill the hole just barely above level; too much filling will leak out the sides when the cookies are sandwiched together. If the cookies do not adhere when placed together, add a little more jam and try again.
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Quickly dunk both sides of an assembled cookie into the purple simple syrup, then place it on a cooling rack or a stack of paper napkins or paper towels to dry slightly (you don’t want to see any liquid dripping off the cookie). Roll in the sugar plum coating to completely cover.
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Finish each Sugar Plum Cookie by inserting a candied stem into the seam of the sandwich. If necessary, use a toothpick to poke a small hole in the seam to make it easier to insert the stem. Serve the cookies at room temperature.
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Storage information: Store any leftover Sugar Plum Cookies at room temperature in an airtight container for up to 1 week; freezing is not recommended.
