What is the correct usage "Last time/The last time"
Solution 1:
Both are the same (and the sentence should have a comma):
"[The] Last time we went to the shop" is an adverbial clause:
"[The] Last time we went to the shop, we bought eggs = We bought eggs on the last occasion that we went to the shop."
In essence, it does not differ from
"Yesterday, we bought eggs" or "We bought eggs yesterday."