expression
how dare you; how could you; how can you have the nerve to
Rhetorical expression of indignation or disbelief, questioning someone's audacity to do something shameful or brazen. Often used when someone shows up or says something after a previous failure or wrongdoing.
あんな失敗をしておいて、どの面下げて戻ってきたんだ。
After such a failure, how dare you come back?
どの面下げて、また金を借りに来たんだ。
How can you have the nerve to come asking for money again?
Describes a person's shameless or impudent character, while どの面下げて is a set phrase directly challenging someone's audacity in a specific situation.
Also expresses indignation ('how dare you'), but よくもまあ is more general and can be used for surprise or admiration, whereas どの面下げて specifically targets the person's shamelessness in showing their face.
Literally 'with what face lowered', from どの (which) + 面 (face) + 下げて (lowering). The image is of someone so ashamed they should lower their face, yet they appear brazenly; the phrase questions how they can show their face without shame.