Line a baking sheet with parchment paper.
Heat 11 tablespoons (142g) butter in 10-inch skillet over medium-high heat until melted, about 2 minutes. Continue cooking, swirling pan constantly until butter is golden brown and fragrant, 4–5 minutes.
Remove skillet from heat and transfer browned butter to large heatproof bowl with remaining 3 tablespoons (58g) butter; using a rubber spatula to scrap butter and solids into bowl. Stir until completely melted; let cool 15 mins.
Meanwhile, whisk together 1 ¼ cups (134g) all-purpose flour, ½ cup (65g) bread flour and ¾ teaspoon (5g) baking soda, ½ teaspoon nutmeg and ¼ teaspoon cardamom in medium bowl; set aside.
Add ¾ cup (150g) dark brown sugar, ½ cup (100g) granulated sugar, 1¼ teaspoons kosher salt, 1 tablespoon vanilla and 1 teaspoon orange extract to bowl with butter and whisk until fully combined.
Add 1 large egg and 1 large egg yolk and whisk vigorously until mixture is light pale in color, fluffy, and thickened to a near-frosting consistency, 3 minutes (you should be able to hold the bowl upside down and the batter not move).
Using a rubber spatula, stir flour mixture into butter mixture and stir constantly for 3 minutes. Stir in ⅔ cup (95g) candied orange and ⅔ cup (93g) halva.
Divide dough into 18 portions, each about 3 tablespoons (or use #24 cookie scoop) and arrange on a baking sheet (no need to space them out yet). If dough is too soft and not holding a firm shape, chill dough balls in refrigerator for 30 minutes before moving on.
Top each dough ball with additional ½ cup candied orange and ½ cup halva. Dip one half of each dough ball in sesame seeds; transfer baking sheet to freezer and chill at least 30 minutes.
Heat oven to 375ºF (190ºC) with rack set in middle position. Line 2 baking sheets with parchment paper.
Arrange 6 frozen dough balls, 3-inches apart, on a prepared sheet. (Smaller baking sheets can be used, but will require multiple batches. Bake 2 cookies at a time on quarter sheet pans).
Transfer sheet to oven and bake until edges are set but centers still look raw, 12 minutes. Lift sheet pan up and tap pan against rack twice, rotate pan and continue baking until edges are golden and centers look underbaked and puffy, 2–3 more minutes.
Remove sheet from oven and tap against stove top once more. Sprinkle with kosher or flaky sea salt and let cool on pan 5–10 minutes. Once cool enough to move, transfer cookies to a wire rack to cool completely.