Ore is a very masculine and considered by some to be a bit arrogant. More neutral is to say boku for male speakers or watashi for either sex. Apparently it’s becoming more popular and neutral under the younger generation, though. It’s also one of those words that is overrepresented in anime, similar to how virtually nobody will say kisama unironically in Japan.
However, actually the most natural is to omit ore wa entirely. Putting it in the sentence makes the speaker the topic of the conversation, and thus the translation is more like “It is I, who saw the steak grilling cook”.
Steeko is close, it’s actually suteeki. Not to be confused suteki, which means “wonderful”.
Kuuku is close too, it’s actually kokku. The etymology leads back to the Dutch word “kok” for cook. There’s a Japanese word ryouri + nin (“cooking” + “person”) for cook as well.
guriru is a noun, used for the (barbecue) grill itself. As far as I can find, it’s not one of those nouns that can become a verb by adding suru. Even if it were, guriru shite no kokku would be ungrammatical. The particle no is used to connect two nouns, and guriru shite is not a noun. Usually a sentence can become a subordinate clause simply by gluing it in front of the noun: guriru shite kokku. But there’s another problem: the -te form of verbs does not form a sentence, it should be guriru suru kokku to have a proper subordinate clause.