For cake: Preheat oven to 350°. Butter and flour three 8-in. round cake pans.
In a large bowl with a mixer on medium-high speed (use the blade or rotary beater if using a standing mixer), beat butter and sugar until fluffy and pale yellow, 4 to 5 minutes. Add eggs 1 at a time, beating well after each addition and scraping down sides of bowl as necessary. Beat in molasses.
In another bowl, whisk flour, baking powder, ginger, cinnamon, salt, and cloves. Stir (or beat at low speed) a third of the flour mixture into butter mixture. Stir in half the milk just until blended. Stir in another third of the flour mixture, then remaining milk, followed by remaining flour, scraping down sides of bowl as necessary. Scrape batter equally into prepared pans and spread level.
Bake cakes until a wooden skewer inserted in the center of a layer comes out clean, 25 to 30 minutes. Cool in pans on racks for 10 minutes, then invert cakes onto racks and remove pans. Cool completely before assembling.
For frosting: In a large bowl, beat butter with a mixer on low speed until light and fluffy. Beat in cream cheese until well incorporated, scraping down sides of bowl as necessary. Beat in powdered sugar, lemon juice, and molasses, then increase speed to medium-high and beat until no lumps remain.
Cut 4 strips of waxed paper or parchment paper, each about 4 by 12 in., and arrange them in a square over edges of a serving platter.
With a serrated knife, carefully slice cake layers in half horizontally. Place first split layer, cut side down, on serving platter (adjust parchment paper so that edges of platter are covered). Spread layer with about 1/2 cup frosting. Sprinkle with about 1/4 cup chopped toffee. Repeat frosting and sprinkling with next 4 split layers. Top with remaining split layer, but don't frost or sprinkle with toffee. Brush away crumbs around sides and base of cake.
Spoon 1 cup frosting into a small bowl and stir in milk until smooth. Spread a thin layer of frosting over top and sides of cake to seal in crumbs. Refrigerate cake until coating is set, about 1 hour. Spread remaining frosting over top and sides of cake, then remove strips of paper. If not serving within a few hours, cover and refrigerate for up to 1 day (see Notes).
Shortly before serving, garnish top of cake with toffee chunks.
Note: Nutritional analysis is per slice.