The correct answer is My brother loves to cook, but he especially loves to barbecue.
Step-by-step explanation:
The sentence "My brother loves to cook, he especially loves to barbecue" is a run-on sentence because each of the sections in the sentence is a complete sentence, and complete/independent sentences cannot be connected using a comma. Instead, a semicolon should be used (;) or a conjunction such as and, but, or can be added after the comma. This second method of correction occurs in the sentence "My brother loves to cook, but he especially loves to barbecue".