Joe in US chasing my dog (5)
I believe the answer is:
corgi
'dog' is the definition.
(I know that corgi is a type of dog)
'joe in us chasing my' is the wordplay.
'joe in us' becomes 'GI' (GI Joe can mean American soldier).
'chasing' means one lot of letters go next to another (some letters chase after others).
'my' becomes 'cor' (both are expressions of surprise).
'gi' after 'cor' is 'CORGI'.
(Other definitions for corgi that I've seen before include "Pre-2018 royal dog" , "Cardigan or Pembroke dog" , "Dog of Welsh breed, royal favourite" , "Cardigan, possibly" , "Welsh 11" .)