Heat the oven to 350°F (175°C). Line a 9x13-inch baking pan with parchment, leaving a 1–2 inch overhang to lift the bars out later, and lightly spray the paper with cooking spray.
Make the base: in a large mixing bowl, beat the softened butter and granulated sugar together on medium speed until pale and a little fluffy, about 2–3 minutes. Add the whole egg and the two egg yolks, vanilla and instant espresso powder, and mix until smooth.
Combine the flour, baking powder and salt in a separate bowl. Add the dry mixture to the butter mixture and stir just until it comes together into a slightly crumbly dough; avoid overmixing. Press the dough evenly into the prepared pan using the back of a spoon or your fingertips to form a compact layer.
Scatter the chocolate chips evenly over the dough, then sprinkle the mini marshmallows on top. Gently press the marshmallows into the chocolate so they sit level with the surface.
Whip the egg whites: in a clean, dry bowl with the whisk attachment, beat the two reserved egg whites on medium-high until they form soft peaks. With the mixer running, add the light brown sugar a few tablespoons at a time and continue beating until the meringue is glossy and holds stiff peaks — the sugar should feel dissolved when you rub a little between your fingers.
Spread the brown sugar meringue over the marshmallows and chocolate in an even layer. Smooth the top with an offset spatula, then shake the pan once or twice to level everything. If the top is browning too quickly during baking, tent loosely with foil halfway through.
Bake until the meringue is golden and set and the edges are lightly browned, about 32–35 minutes. Remove from the oven and allow the pan to cool on a wire rack until the bars reach room temperature.
Chill for 20–30 minutes in the refrigerator to make slicing cleaner, then use the parchment overhang to lift the slab from the pan and cut into 24 squares. Store in an airtight container; separate layers with wax paper to protect the meringue.