Combine the all-purpose flour, durum flour, semolina flour, salt, and sugar in a food processor and pulse to combine. Add the butter and pulse just until most of the butter has coated the flour and the largest remaining pieces are no bigger than peas. The mixture will look slightly darker and crumbly.

With the processor running on pulses, slowly pour the ice water through the feeder tube 1 to 2 tablespoons at a time. After about ¼ cup, squeeze a small amount of dough — it should just hold together. Continue adding water until it holds; you will likely use about ½ cup total. Stop while the dough still looks crumbly but cohesive. Do not over-process.

Turn the dough out onto the counter and press with the heel of your hand just until it comes together. Add a few drops of ice water only if it keeps crumbling. Form into a disk, wrap in plastic, and refrigerate for at least 2 hours or overnight. The dough freezes for up to 3 months.

Roll the chilled dough on a semolina-dusted surface to about ⅛ inch thick and 12 to 14 inches across. Slide onto a parchment-lined sheet pan and refrigerate while you prepare the frangipane.

Make the almond cream: beat the butter and sugar together until smooth, then mix in the almond flour, eggs, and cornstarch until fully combined. Set aside at room temperature.

Melt the chocolate and let it cool until fluid but no warmer than body temperature. Allow the frangipane to reach room temperature as well. Add the Dutch-process cocoa and melted chocolate to the frangipane and stir with a stiff silicone spatula until completely smooth and no streaks remain.

Preheat the oven to 350°F conventional (no fan).

Spread all of the chocolate frangipane over the chilled crust, leaving a 1½-inch border bare. Scatter the diced strawberries over the frangipane, then lay the sliced strawberries on top in a random pattern, flat and not overlapping enough to bury the filling.

Fold the bare border up and over the edge of the fruit, pleating as you go. Refrigerate the assembled galette for 20 to 30 minutes so the border holds its shape.

Brush the border with cream. Sprinkle the entire galette, filling and all, with turbinado or white sparkling sugar.

Bake at 350°F for 40 to 45 minutes, until the crust is deep golden and the frangipane feels set when pressed lightly at the center. The filling will look matte rather than wet where it peeks through the strawberries, and the fruit will be bubbling. Do not rely on color alone — the chocolate filling obscures the usual visual cues.

Cool on the pan before sliding off the parchment.