Preheat oven to 350°F. Stir 3 Tbsp. melted unsalted butter into 1 1/3 cups chocolate cookie crumbs; press into a 9-inch tart pan. Bake 5 minutes.
Beat 1/4 cup cornstarch, 1/2 cup sugar and 2 large eggs until smooth. Heat 2 cups milk and a pinch of salt until it simmers, whisking. Ladle 1/3 of milk into egg mixture, whisking, then pour back into pan. Lower heat; cook, whisking, until thickened. Strain and whisk in 3 oz. chopped bittersweet chocolate, stirring until melted. Cool 5 minutes; whisk in 2 Tbsp. unsalted butter. Pour into crust; chill at least 2 hours or overnight.
In a pan, cook 1/3 cup sugar and 1/2 cup water, swirling pan occasionally, until mixture turns light amber. Add 1 lb. halved, pitted bing cherries; cook, stirring, for 5 minutes. Remove cherries from liquid. Mix 2 tsp. cornstarch with 1 Tbsp. water and add to cherry liquid; cook for 2 minutes. Transfer cherries and liquid to bowl and chill. Spoon cold compote over tart.