For the cookies: Preheat the oven to 375 degrees F. Line 2 large rimmed baking sheets with parchment paper.
Whisk the flour, baking powder and salt in a small bowl.
Stir the 2 cups chocolate chips and butter in a medium bowl over a saucepan of barely simmering water until the chocolate mixture is melted and smooth. Remove the bowl and set aside. The mixture may be thick.
Combine the eggs, sugar and vanilla in a large bowl. Beat the egg mixture using an electric mixture, until light and fluffy, about 5 minutes. Add the melted chocolate mixture and beat until well blended. Add the flour mixture and beat until just incorporated. Stir the remaining 1 cup chocolate chips into the batter. If the batter is very soft, chill for 15 to 20 minutes to allow the batter to firm up slightly. The batter can be prepared 1 day ahead. Let the batter stand at room temperature at least 1 hour before continuing.
Using a small cookie scoop, drop the batter in rounded tablespoonfuls onto the baking sheets, spacing 1 1/2-to-2-inches apart. Bake the cookies until slightly firm to touch and crackled all over tops but still soft in the center, 12 to 15 minutes. Cool the cookies on the sheets.
For the white chocolate ganache: Bring the cream to simmer over medium heat in a heavy small saucepan. Remove the pan from heat and add the white chocolate chips. Stir until melted and smooth. Chill the ganache until firm enough to spread, 1 to 2 hours. The ganache can be prepared 1 day ahead. Cover and keep chilled. Let the ganache stand at room temperature 2 hours to soften slightly before using.
For the strawberry preserves: Mix the preserves and the lemon zest in a small bowl.
Spread about 1 teaspoon ganache onto the flat side of each cookie to within 1/4-inch of the edge. Spoon 1/2 to 1 teaspoon strawberry preserves on the ganache side of half of the cookies. Place the remaining cookies on top, ganache side down, to form sandwich cookies. Chill until the ganache is firm, 15 to 20 minutes. Insert lollipop sticks into the ganache filling, if desired
For the white chocolate drizzle: Place the white chocolate in a small bowl over a saucepan of barely simmering water. Stir occasionally until the chocolate is melted and smooth. Drizzle the melted chocolate over the top of one side of each sandwich cookie, using a fork. Chill until the chocolate has set, about 10 minutes. The drizzle can be prepared 1 day ahead. Store airtight in the refrigerator.