[{"Contact":[{"name":"ABC","phone":5501234,"mail":"abc@mail.com"},{"name":"DEF","phone":6001234,"mail":"def@mail.com"}],"nomatter":"trash"}]

db.collection.aggregate([{$match:{"Contact.name":"ABC"}},{$project:{Contact:{$filter:{input:"$Contact",cond:{$eq:["$$this.name","ABC"]}}}}},{"$replaceWith":{mail:{$first:"$Contact.mail"}}}])