In a small microwave-safe bowl, melt 8 ounces of chocolate in 30 second intervals in the microwave until smooth. Dip the top of each cone into the chocolate so there is a 1-inch rim, then roll in the crushed or ground espresso. Place the dipped cones on a wax paper-lined plate and refrigerate. When firm, about 20 minutes, remove from the refrigerator and set cones, dipped side up, inside small glasses so they stand upright. Spoon the remaining melted chocolate evenly into the sugar cones and return (glass and all) to the refrigerator.
Meanwhile, hull and quarter 2 pints of strawberries and puree them in a food processor with 2 tablespoons of the lemon juice, 5 tablespoons of the sugar and a pinch of salt. Strain the puree through a fine-mesh strainer into a small saucepan. Bring to a simmer over medium-high heat, and cook until reduced by half, about 35 minutes, stirring frequently. Pour into a small bowl and chill the puree until cold.
In the bowl of a stand mixer fitted with a whisk attachment, whip the heavy cream and mascarpone to stiff peaks and then fold in the puree in 2 batches. Refrigerate until firm and set, 1 to 2 hours.
Just before serving, hull the remaining 1 pint of strawberries and cut them lengthwise into 1/4-inch slices. Toss with the remaining 1 tablespoon of lemon juice and 2 tablespoons of sugar.
To assemble: Spoon strawberries into the cones until they are 1/4 full. Top that with approximately 1/4 cup of the mascarpone mixture, using an ice cream scoop that has been dipped in warm water. Garnish with shaved bittersweet chocolate and refrigerate until ready to serve.