Here are two suggestions for you:
You need to add your admin account in to "Discovery Management" permission group first:
Suggestion 1: Search-Mailbox command:
You can use command below to check whether could find correct email first(Search in User1 mailbox to find emails that sent to user2):
Search-Mailbox -Identity User1 -SearchQuery "sent>=06/01/2020 AND sent<=03/08/2021 -and recipients:******@domain.com" -EstimateResultOnly
If this command running successfully, you could export the result into the mailbox that you want:
Search-Mailbox -Identity User1 -SearchQuery "sent>=06/01/2020 AND sent<=03/08/2021 -and recipients:******@domain.com" -TargetFolder "result" -TargetMailbox "******@domain.com"
You will could find those emails in the "check" mailbox.
Suggestion 2: Use in-place eDiscovery & hold in Exchange admin center:
After searching complete, you could download the result:
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