This is a workaround for Twitter search API being limited to only tweets from the past week.
Open Twitter on your favourite browser, open dev-tools, then go to console.
Make sure you are on Twitter search page, searching for your mentioned handle or hashtag or whatever.
Enter the code, it will only capture the handles but you can alter it to capture more data of your choosing.
It will attempt to adjust the page scroll to reach the bottom, in order to load more tweets, until you reach the end.