Make the cake
Position racks in the bottom and top thirds of the oven and heat the oven to 350°F.
Butter three 9x2-inch round cake pans and line each with a parchment round. Butter the parchment.
In a large bowl, stir the flour, baking powder, and salt.
Using stand mixer fitted with the paddle attachment, cream the butter and sugar on medium high until light and fluffy, about 3 minutes. Add the eggs one at a time, beating for 30 seconds after each addition.
On low speed, add the flour mixture to the butter mixture in three increments, alternating with the milk in two increments, starting and ending with the flour. Beat on low speed between each addition until just incorporated. The batter will be thick and fluffy. Stir in the pears, candied ginger, and vanilla.
Divide the batter evenly among the three prepared pans. Level the batter with a spatula. Set two pans on the top rack and the third on the lower rack. Stagger the pans on the oven racks so that no pan is directly over another. Bake, swapping and rotating the pans’ positions after 20 minutes, until a toothpick inserted in the center of each cake comes out clean, 40 to 45 minutes. Cool on racks for 30 minutes. Invert the cakes onto the racks, remove the parchment, and cool completely.
Make the caramel glaze
Fill a cup measure with water and put a pastry brush in it. In a heavy-duty 3-quart saucepan, stir the sugar, lemon juice, and 1/3 cup cold water. Brush down the sides of the pan to wash away any sugar crystals. Bring to a boil over medium-high heat and cook, occasionally brushing down the sides of the pan, until the mixture starts to color around the edges, 5 to 8 minutes. Gently swirl the pan once to even out the color and prevent the sugar from burning in isolated spots so that the sugar caramelizes evenly. Brush down the sides of the pan if the sugar threatens to burn. Continue to cook until the sugar turns medium amber, about 30 seconds more. (Once the mixture begins to color, it will darken very quickly so keep an eye on it.)
Reduce the heat to low and carefully stir in the butter. It will bubble up. Keep stirring until the bubbles settle down, then add the cream a couple of tablespoons at a time, stirring to combine after each addition. Continue stirring until the caramel is smooth, about 2 minutes.
Remove from the heat and stir in the rum and salt. Strain the caramel through a fine sieve into a heatproof measuring cup to remove any small pieces of sugar. Cool to room temperature, stirring occasionally. You should have about 1-2/3 cups.
Make the whipped cream
Using a stand mixer fitted with the whisk attachment and a cold bowl, whip the cream on medium speed until it starts to thicken, about 2 minutes. Add the sugar, raise the speed to medium high, and continue to whip until soft peaks form, about 2 minutes more. Raise the speed to high until the cream forms stiff peaks and is very thick, about 30 seconds more.
Assemble the cake
Put a cake layer on a flat serving platter or a cake stand lined with strips of waxed paper to keep it clean while icing. Top the layer with 2 cups of the whipped cream, spreading it evenly with a metal cake spatula almost to the cake’s edge. Put the next cake layer over the filling and spread 2 more cups of the whipped cream over it. Top with the last cake layer and thickly coat the sides and top of the cake with the remaining whipped cream. Refrigerate the cake for 30 minutes.
Glaze the cake
Slowly pour the glaze over the top of the cake, making sure that it covers the top and drizzles down the sides (some of the whipped cream frosting should show through the drizzles on the sides).
Press the coarsely chopped candied ginger pieces in a 1-inch band around the base of the cake for decoration. Remove the waxed paper strips.
Refrigerate the cake for at least 30 minutes and up to 2 hours before serving.
Photo: Scott Phillips