Combine chopped apricots, 3 cups sugar, half-and-half, and salt in a medium saucepan. Cook over medium heat, stirring until sugar dissolves. Cover and continue to cook over medium heat 2 to 3 minutes to wash down sugar crystals from sides of pan. Uncover and cook, without stirring, until mixture reaches soft ball stage (236°). Remove from heat; let cool to lukewarm (110°). Add walnuts and vanilla. Beat with a wooden spoon until mixture stiffens.
Turn mixture out onto a lightly buttered smooth surface; knead until mixture is smooth and opaque. Divide candy into 3 portions; shape each portion into 6-inch rolls. Dust each roll with powdered sugar. Wrap rolls individually in waxed paper; chill overnight. Cut into 1/4-inchthick slices to serve.