Preheat oven to 300 degrees F (150 degrees C). Grease a 9-inch cake pan with butter, and dust with bread crumbs; set aside.
Melt butter in a small saucepan over medium heat. Add the sliced apricots, and drizzle with honey. Cover, and cook gently until the apricots have softened, then allow to cool to room temperature.
Meanwhile, beat the egg whites to soft peaks with an electric mixer. Slowly pour in sugar while continuing to beat to stiff peaks; set aside.
Beat the egg yolks on low speed for a few seconds until slightly thickened.
Slowly pour in sugar while continuing to beat until thickened and light-colored. Beat in milk, then sift together flour with baking powder, and fold into yolks along with cooled apricot mixture, walnuts, and egg whites until combined.
Pour into prepared pan, and bake in the middle or bottom of the preheated oven until a toothpick inserted into the center comes out clean, about 50 minutes. Loosen the cake by running a knife around the edges of the pan.
Invert cake onto a plate, then place another plate on the cake bottom, and flip right-side-up; cool completely before serving.