宛てて書く is not an idiomatic/compound verb. It's just two verbs, connected using the te-form.

宛てて is the te-form of the verb 宛てる (to address (a letter)), and this kind of te-form describes how or under what circumstances the second verb (書く) is done. In general, this type of `V1 + て/で + V2` can be translated as "V2 (while) doing V1" or "V2 by doing V1".

Examples:

> - 歩いて学校に行く go to school by walking / go to school on foot
- 包丁を使ってたまねぎを切る cut an onion using a knife
- 未来の自分に宛てて手紙を書く write a letter addressing the future self
- 注意して歩く walk while paying attention / walk with attention
- 遊んで過ごす live an idle life

See: https://japanese.stackexchange.com/q/38769/5010