At first, pet Alsatian regularly rejected carb-heavy food (5)
I believe the answer is:
pasta
'food' is the definition.
(I know that pasta is a type of dish)
'at first pet alsatian regularly rejected' is the wordplay.
'at' means one lot of letters go next to another.
'first pet' becomes 'p' (1st letter of 'pet').
'regularly rejected' means one should take alternating letters (letters removed at regular intervals).
The alternate letters of 'alsatian' are 'asta'.
'p' put next to 'asta' is 'PASTA'.
'carb-heavy' acts as a link.
This may not be right. It may belong to another bit of the clue.
Can you help me to learn more?
(Other definitions for pasta that I've seen before include "Macaroni, for example" , "Fare from Italy" , "Staple food of Italy" , "Spaghetti, ziti, fusilli, for example" , "Fusilli, tortelloni etc" .)