Make the cookie dough first so it can chill while you make the mochi later.
In a medium bowl, mix both sugars and the butter until the mixtures comes together and becomes lighter in color. This takes some muscle, so you can use a hand mixer or stand mixer if you can’t or don’t want to mix by hand.
Add the egg, and mix until combined.
Add the flour, salt and baking soda. Mix until combined. Refrigerate for an hour.
After your cookie dough has chilled for 30 minutes, start on the mochi. Mix the first 6 ingredients together, and add food coloring to reach your desired color if you’d like (I added 6 drops of red food coloring).
Cover the bowl with a microwavable lid or plastic wrap, and microwave for 30 seconds at a time, for about 2 minutes total. Stir each time you check on the mochi. It’ll be done when it’s thick and no longer liquid-y. Don’t worry if it’s not cooking evenly or looks a little lumpy.
Mix in the butter until the mochi is smooth. Cool until slightly warm or room temperature.
Portion both the cookie dough and the mochi into 20 pieces each. For each piece of cookie dough, roll into a ball, then flatten into a circle with your palms. Place a piece of mochi in the middle, and fold the cookie dough over and around the mochi to make sure the mochi is sealed inside. You should not see any mochi peeking out. Roll the stuffed cookie into a ball if it’s gotten misshapen. Repeat with the remaining dough and mochi.
Place the cookies in the refrigerator. Preheat the oven to 350F.
Prepare 2 baking sheets by placing a silicone baking mat or parchment paper on the sheet. When the oven is ready, space out 6 cookies on each baking sheet (keeping the other dough balls you’re not baking yet in the fridge). Bake for 12 minutes.
The cookies will be puffed in the middle and pretty soft – this is fine. They’ll flatten and firm up as they cool. If you’re using sprinkles, press them into the cookies while they’re still warm and soft. Let cool on the baking sheet for a few minutes before transferring to a wire cooling rack.
Let the baking sheets cool until you can handle them without oven mitts, and bake off the rest of the cookies.
If you’re not eating all of the cookies on the day of baking, either refrigerate or freeze the cookies. Do not leave at room temperature overnight.
I recommend freezing them, even if you’ll be eating them within a few days, since this is the best way to preserve the cookies to as close as fresh baked. You can stack them and freeze. They’re best eaten within a month.
If you plan to eat the cookies within 2 or 3 days, you can refrigerate them, but the mochi will get sticky in the middle, so either separate any layers of cookies with parchment paper or keep the cookies in a single layer.
You need to re-heat the cookies after refrigerating or freezing them. Pop them in an oven (I like using the toaster oven for this) at 350F for 3-5 minutes (3 minutes if you refrigerated them, 5 minutes if you froze them). Let cool slightly before eating.