It looks like the nice link at the top of commits that indicates which PR they're associated with comes from an internal API that isn't exposed to the general public. So far as I can tell the way to do this (using the official API and not reverse engineering any internal endpoints) would be to list pull requests (https://developer.github.com/v3/pulls/#list-pull-requests), filter for ones with linked issues, then grab the merge commit (looks like your repo uses merge commits) to determine which PRs have commits in the desired range.
Here's an example of how to do that. I didn't include how to get commits for a range, just wanted to demonstrate the github api pieces needed. This script prints out the merge commits with associated issues for a given repo as well as the linked issue.
#!/bin/bash
echo "[" > allResults
i=1
numResults=1
firstRun=1
while [[ $numResults -gt 0 ]]
do
curl -X GET -u ${GITHUB_USERNAME}:${HOMEBREW_GITHUB_API_TOKEN} "https://api.github.com/repos/microsoft/msquic/pulls?state=closed&page=$i" > page
numResults=`cat page | jq '. | length'`
i=$[$i + 1]
if [[ $numResults -gt 0 ]]
then
# bit of nonsense to deal with trailing comma problem
if [[ $firstRun -ne 1 ]]
then
echo "," >> allResults
fi
if [[ $firstRun -eq 1 ]]
then
firstRun=0
fi
cat page | awk 'NR != 1 { print buffer; buffer = $0 }' >> allResults
fi
done
echo "]" >> allResults
echo "[" > prsWithIssues
# select only PRs with issues then use the crazy awk business to put
# back the commas in the array
cat allResults | jq '.[] | select(.has_issues == true)' | awk '/^{/ { if (endsWithBrace) {print","$0} else {print} ; endsWithBrace=0} !/^{/ {endsWithBrace=1; print} /.*\}$/ { endsWithBrace = 1 }' >> prsWithIssues
echo "]" >> prsWithIssues
cat prsWithIssues | jq '.[] | "\(.merge_commit_sha) \(._links.issue.href)"' > mergeCommitToIssueLink
# just printing here, you'll probably want
# to do a grep for commits in the range you want
cat mergeCommitToIssueLink