Reformulating a rolled cookie recipe to reduce distortion of design

I want to make homemade oreo-type cookies using an embossed roller with a recipe I've used and love, but it puffs too much to keep the detail. Brave tart's oreo cookies are the look I'm after, but her recipe has far less cocoa and chocolate. Is it possible to change this recipe to avoid spreading and puffing?

Here's the recipe:

Homemade Oreo Cookies
Source: Flour Bakery, via Boston Globe

1 cup (2 sticks/228 grams) unsalted butter, melted and cooled slightly
¾ cup (150 grams) granulated sugar
1 teaspoon vanilla extract
1 cup (200 grams) semisweet chocolate chips, melted and cooled slightly
1 egg
1½ cups (210 grams) unbleached all-purpose flour
¾ cup (90 grams) Dutch-processed cocoa powder
1 teaspoon kosher salt
½ teaspoon baking soda

1. In a medium bowl, whisk together the butter and granulated sugar until well combined. Whisk in the vanilla and chocolate. Add the egg and whisk until thoroughly incorporated.
2. In another medium bowl, stir together the flour, cocoa powder, salt, and baking soda until well mixed. Using a wooden spoon, stir the flour mixture into the chocolate mixture. The dough will start to seem too floury, and you will find it easiest to switch to mixing it with your hands until it comes together. It will have the consistency of Play-Doh. Let the dough sit at room temperature for about 1 hour to firm up.
3. Transfer the dough to a 15-inch square sheet of parchment or waxed paper. Using your hands, shape the dough into a rough log about 10 inches long and 21/2 inches in diameter. Place the log at the edge of the sheet of parchment paper, and roll the parchment around the log. With the log fully encased in parchment, roll it into a smoother log, keeping it at 21/2 inches in diameter. Refrigerate for at least 2 hours, or until firm. The log may settle and sink a bit in the fridge, so reroll it every 15 minutes or so to maintain a nice round log, if you like. If not, your cookies will be more oblong than round, which is not a bad thing taste-wise, though they won’t look like the famous packaged cookie. (At this point, the dough log can be well wrapped in plastic wrap and stored in the refrigerator for up to 1 week or in the freezer for up to 1 month. If the dough is frozen, thaw overnight in the refrigerator before proceeding.)
4. Position a rack in the center of the oven, and heat the oven to 325 degrees. Line a baking sheet with parchment paper or butter it.
5. Cut the dough log into 1/4-inch-thick slices. Place the slices about 1 inch apart on the prepared baking sheet.
6. Bake for 20 to 25 minutes, or until the cookies are firm to the touch. Check them frequently after 16 or 17 minutes, poking them in the middle. As soon as they feel firm to the touch, remove them from the oven. You can’t judge by color because they start out black. Let cool on the baking sheet on a wire rack to warm or room temperature.

For reference, here's Brave Tart's recipe, via Serious Eats:

Ingredients
For the Chocolate Wafers:
4 ounces unsalted butter (about 8 tablespoons; 115g), creamy and soft, about 68°F (20°C)
3 1/2 ounces sugar (about 1/2 cup; 100g)
2 ounces golden syrup (about 3 tablespoons; 55g), such as Lyle’s
1/2 teaspoon baking soda
1/4 teaspoon (1g) Diamond Crystal kosher salt; for table salt, use about half as much by volume or the same weight
1/4 teaspoon coconut extract (optional)
5 3/4 ounces bleached all-purpose flour (about 1 1/4 cups, spooned; 165g), such as Gold Medal
1 1/4 ounces Dutch-process cocoa powder (about 1/3 cup plus 1 tablespoon; 35g), such as Cacao Barry Extra Brute, plus more for dusting
For the Filling:
6 ounces unsalted butter (about 12 tablespoons; 170g)
1 teaspoon vanilla extract
1/8 teaspoon Diamond Crystal kosher salt
8 1/2 ounces powdered sugar (about 2 cups plus 1 tablespoon; 240g)

Directions

For the Chocolate Wafer Dough: Combine butter, sugar, golden syrup, baking soda, salt, and coconut extract (if using) in the bowl of a stand mixer fitted with a paddle attachment. Mix on low speed to moisten, then increase to medium and beat until fluffy and light, about 5 minutes, pausing to scrape bowl and beater about halfway through.

Sift flour and cocoa together. (If using cup measures, spoon into the cups and level with a knife before sifting.) With mixer running on low speed, sprinkle flour/cocoa mixture into butter mixture. It will seem dry and mealy at first, but continue mixing to form a smooth dough. Knead against sides of bowl to form a smooth ball, then divide in half and flatten into disks. Use immediately or wrap in plastic and refrigerate up to 1 week; soften cold dough 30 minutes at room temperature, then knead on a bare surface until pliable and smooth.

For the Wafers: Adjust oven rack to middle position and preheat to 350°F (180°C). On a cocoa-dusted surface, roll a portion of dough into a 7-inch square. Sprinkle both sides with cocoa and roll until 1/4 inch thick; generously dust with cocoa and continue rolling to 1/8 inch (see note). Alternatively, roll dough to 1/8 inch using an embossed pin. Slide an offset spatula under dough to loosen, brush away excess cocoa, and stamp into 1 1/2–inch rounds.

Arrange on a parchment-lined aluminum baking sheet, leaving 1/4 inch between wafers. Gather scraps, knead, re-roll, and cut as before. Repeat with remaining dough as well. Any remaining scraps can be discarded or baked to grind for crumbs. Bake until wafers are firm and dry, about 15 minutes, and cool to room temperature on the baking sheet. Fill immediately or store in an airtight container up to 1 week at room temperature.