A lil' extravagant
Signifying a barbecue. A chef hat element could give a look for the professional kitchen. Has chips and firied chicken and fried fish to reflect the food you serve ---at least main dishes -- as you have asked. The first two images have a tick mark on the chicken - subtle -- but I sort of liked it and so throwing it at you. The rest are sort of badge style logos you could possibly use for badges or for different staff uniform, either having the background color on the cloth and the logo on it or a neutral color with the badge style logo printed on it. Grayscale ones in the tail end which could go on possibly coffee mugs if you serve coffee or on the packaging to go --- with a twist of the red tick mark on the grayscale one for your consideration. Finally a grayscale version of the badge style design. With regard to the website, please message me on your idea of implementation and I can give you a quote for it. Please send feedback on chat if you like this idea & would want any changes.