Put the apple jam and egg white in a mixing bowl and whip until very stiff, starting at a low speed of the food processor and gradually increasing it to maximum.
Cook the sugar syrup in parallel. (If you use a hand mixer, first beat the apple and egg white mixture until stiff and then cook the sugar syrup). To do this, place the sugar, agar-agar, and water in a small saucepan with high sides and a thick bottom, bring to a boil while stirring constantly, and then simmer over low heat, stirring, until the sugar syrup reaches 110°C.
Once the sugar syrup has reached the required temperature, slowly add it in a thin stream to the stiffly whipped apple and egg white mixture, without stopping to continue beating it. Then whip the mixture again for about 3 - 5 minutes until very stiff.
Fill the zefir mixture into the piping bag with the star nozzle, pipe rosettes of it on a sheet of baking paper (I made each rosette in two layers.) and let them dry at room temperature for about 12 hours, until the zefir becomes firm on the outside and inside.
Take two rosettes at a time and glue them together with the underside, which should be a little sticky. Roll the finished zefir in powdered sugar, shaking off the excess powdered sugar.