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To make the ube cookie dough: In a large bowl or the bowl of a stand mixer, beat (“cream”) together the butter and sugar until pale in color and fluffy, 3 to 4 minutes. Scrape the sides of the bowl as needed.
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To the butter/sugar mixture, add the egg, jam, and extract and mix until incorporated, 1 to 2 minutes. Add the flour, baking powder, and salt and mix until just combined, less than 30 seconds. Cover and refrigerate for at least 1 hour or overnight.
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To make the mochi filling: In a medium microwave-safe bowl, whisk the rice flour, confectioners’ sugar, cornstarch, and water until no lumps remain. Cover loosely, microwave on high power for 1 minute, then use a spatula to mix until evenly combined. Microwave for another minute and mix again. If there are any wet parts remaining, microwave for an additional minute at 50% power. The mochi is ready when it's slightly translucent, cohesive, and evenly cooked.
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Add the butter and mix thoroughly to incorporate.
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Cover and let cool until the mochi is no longer too hot to touch, about 8 to 10 minutes.
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Working with damp hands, stretch and fold the mochi until it's smooth, elastic, and homogenous, about 15 seconds. It’ll still be sticky. Cover and set aside until ready to bake.
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Preheat the oven to 350°F and place a rack in the center. Line a baking sheet with parchment.
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To assemble the ube cookies: Working with damp hands, pull off 12 tablespoon-sized pieces of mochi (about 10g each) and place them on a nonstick surface, like a silicone mat or piece of plastic wrap, as you work. If the mochi is too sticky to handle, rub a bit of neutral oil on your palms. (You will have a bit of leftover mochi; see "tips" below for details on delicious ways to use it up.)
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Scoop 1 1/2 tablespoons (about 35g) of cookie dough onto the prepared sheet; a heaping tablespoon cookie scoop works well here. Shape the scooped dough into balls. Pat and press one ball of cookie dough into a 3" round between your palms, lightly flouring your hands as needed to prevent sticking. Place a mochi ball in the center of the disk, wrap the cookie dough around it, close the seams, and re-roll into a smooth, firm ball. Distribute the shaped cookies evenly on the prepared baking sheet, leaving about 2" of space between them.
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Place the granulated sugar and confectioners’ sugar each in their own a shallow bowl. Working with one dough ball at a time, toss it in the granulated sugar, coat thoroughly, then transfer it to the confectioners’ sugar and turn until well-coated. (Coating in granulated sugar first prevents the coating from dissolving into the dough.) Place on the prepared baking sheet and repeat until you’ve made a total of 12 dough balls.
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Bake the cookies for about 16 to 18 minutes, until the edges have set. The cookie centers should be soft but not wet or shiny.
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Cool on the baking sheet for a few minutes, then transfer to a wire rack to cool completely.
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Storage instructions: Store ube cookies loosely wrapped at room temperature for up to 3 days (the mochi will a get bit firmer over time). Freeze mochi-stuffed cookie dough balls after step 9 (assembly) and before step 10 (coating in sugar). When ready to bake, let dough balls sit at room temperature for 5 to 10 minutes before coating in sugar (this will make the sugar adhere more readily). Bake as directed, adding 4 to 5 minutes to the bake time if necessary. Baked cookies can also be frozen. Thaw in the refrigerator overnight before enjoying.
