Okay, some tries later..
You can put the "my" at the end of the string list and add the asterisk to search for thread titles (as you wanted) with it on last position.
You can as well add the "dogs" beforehand. But it seems to only search for it, if it's actually tagged with the label "dogs", which not many users do and as such the tags are quite low in usage. Or there are no threads labeled by "my" and "dogs" which would surprise me a bit.
As example if you search for "mare my*" in only titles, then it finds one result which has the tag "mare", with "pony my*" it finds it as well. In both cases reliable regarding the "my"-aspect, even if not tagged.
As the search string "girl my*" proves, it works like intended for some cases where there's actually characters following up the my.. - but I am not sure it still works with spaces. This string finds exactly two results, if searched only in titles.
Can you give me the links to the threads you want to find? Maybe that helps further than searching for something without knowing what it is.
Edit:
Okay, solution found.
Even as the search mask tells that it ignores the "my" in this case, it still includes it and puts out almost exactly what you wanted.
Use "my?dogs"
WITH the " " in the mask.
Activate the checkmark on "search titles only".
Looks like this, then:
If you want to search for the string: "Its in me ok"..
..use "Its?in?me?ok" in the mask, together with the ".
But I did not find any results, probably because this title doesn't exist.