Preheat your oven to 300°F. In a bowl, combine the 1/4 cup sugar, 1 tbsp brown sugar, 1/2 cup flour, 1/2 tsp baking powder, 4 tbsp canola oil, and 1 tsp vanilla extract, mixing until the mixture resembles coarse crumbs. Spread the mixture on a baking tray and bake for 15-18 minutes until golden and fragrant, stirring halfway through. Remove from oven and let cool completely—this takes about 10-15 minutes. Once cooled, crumble any larger pieces so you have about 1/2 cup of crumbs for the cookies and reserve the rest for topping.
While the crumbs bake, finely dice the strawberries into 1/4-inch pieces and place them in a small bowl. Toss with the lemon juice to coat evenly—the lemon juice prevents browning and brightens the strawberry flavor. Set aside until ready to use.
In a large bowl, beat 12 tbsp unsalted butter with 1 cup brown sugar and 1/4 cup sugar for about 3 minutes until light, fluffy, and pale in color. This creaming process incorporates air into the dough, which helps the cookies rise and develop a tender crumb. Stop and scrape down the bowl halfway through to ensure even mixing.
Add the large egg and egg yolk (make sure they're at room temperature for smooth incorporation) to the creamed butter mixture, followed by 2 tsp vanilla extract. Beat until well combined and smooth, about 1 minute. I like to scrape the bowl between additions to ensure everything is fully incorporated and there are no streaks of egg white.
In a separate bowl, whisk together 2 1/4 cups flour, 2 tsp cornstarch, 1 1/4 tsp salt, 1 tsp baking soda, and 1/2 tsp baking powder. This prevents lumps and distributes the leavening agents evenly. Add the dry mixture to the butter-egg mixture and stir until just combined. Fold in the 1/2 cup shortcake crumbs from Step 1 and the strawberry mixture from Step 2 gently—don't overmix, as this keeps the cookies tender and prevents the strawberries from breaking apart.
Scoop the dough onto parchment-lined baking trays using a cookie scoop or spoon, spacing them about 2 inches apart. Gently press some of the remaining shortcake crumbs onto the top of each cookie. Freeze the shaped dough for 2-3 hours or up to overnight. Chilling is essential here—it prevents the cookies from spreading too much during baking and helps them keep their shape. I find that freezing also allows the flavors to meld together for a better-tasting cookie.
Preheat your oven to 350°F and bake the frozen cookies directly from the freezer for 9-11 minutes for smaller cookies or 12-14 minutes for larger cookies, until the edges are set but the centers still look slightly underdone. Let the cookies cool on the baking tray for 5 minutes, then transfer to a wire rack to cool completely for about 15 minutes. This cooling time allows them to firm up while staying tender inside.