Building onto or next to the house
Convenience is the biggest advantage of this scenario, because it will be much simpler (and less expensive) to run water, electricity, and natural gas out to the outdoor kitchen. Transporting food outside will also be easier. When the outdoor kitchen is attached to the house, you can make a pass-through, or use a window as a pass-through, adding even more convenience. In one kitchen I designed, we built a window pass-through as well as an interior underwindow bookcase with a wide ledge at the top covered with zinc sheeting so that it functions as a serving place in the indoor kitchen and a landing place when things are going out to and coming in from the outdoor kitchen.
Another great advantage of an attached outdoor kitchen is that the outdoor space is buffered from weather by the walls of the building. An outdoor kitchen in an ell formed by two wings of a building or in a courtyard protected on three sides by the house, or even an outdoor kitchen that only touches the house on one side, will have a distinct advantage over a freestanding structure: Winds will be considerably less fierce, rain and snow may infiltrate less, and the wear of weather on the kitchen elements may be moderated.
On the other hand, an attached outdoor kitchen may not really feel like the outdoors, because it occupies a space so close to the house. Also, because of its proximity to the doors and windows of your home, it may require the installation of an exhaust fan to direct heat and smoke away from the building.
Finally, although it is always important to design your outdoor kitchen with the style of your home in mind, it is even more crucial when it is actually attached to the house. Keep the scale consistent, choose similar materials and style elements, and use the roofline of the house as your guide when designing a roof for the outdoor kitchen.