Cheese doesn't need a bonfire.
Smoke works beautifully with firm cheese. A little apple or alder against a decent mature cheddar, and suddenly a very ordinary sandwich has ideas above its station. A lot of smoke, however, can make cheese taste as though you've scraped it off the inside of a railway tunnel.
The catch is that cheese melts at temperatures far below the conditions used for cooking meat. People call the answer 'cold smoking', which is perfectly reasonable as a flavour description and a terrible invitation to ignore food safety. Smoke doesn't cook cheese, sterilise it or preserve it.
Start with cheese that can take it.
Choose firm cheese made from pasteurised milk, bought chilled from a reputable supplier, in its use-by date: mature cheddar, gouda, hard alpine-style cheeses or a firm mozzarella intended for cooking. Avoid using very soft cheeses, mould-ripened cheeses or raw-milk cheeses for casual home-smoking experiments. Some are more susceptible to contamination and deserve tighter control.
Cut 300–500 g / 11–18 oz into relatively generous blocks. Small cubes take smoke rapidly and can become unpleasantly intense. Keep the cheese well chilled until the moment it's being flavoured.
The practical home approach: brief smoke, not a long outdoor cure.
The most straightforward low-risk approach is a purpose-made culinary smoking gun used with food-safe hardwood chips and the maker's instructions. Put chilled cheese on a clean rack in a food-safe, heat-resistant lidded container specified for the equipment. Introduce a brief charge of cool smoke without placing hot embers in contact with the cheese. Let it settle for the short period suggested by your appliance, typically minutes rather than hours, then return the cheese promptly to the refrigerator.
This is a flavour infusion, not a shelf-life extension. Keep handling time out of refrigeration as short as possible and make sure the cheese remains safely chilled; don't treat an unmeasured outdoor smoker on a sunny day as a cold room. Taste a small piece after refrigeration. If it needs more smoke, try another short session rather than ruining the whole block in one go.
Do not light wood or charcoal inside a refrigerator, kitchen cupboard or enclosed room. Use smoking equipment outdoors or in a location expressly approved by the manufacturer's instructions, with proper ventilation and fire precautions. Many smoke generators produce heat as well as smoke; don't assume the word 'cold' guarantees safe product temperature.
What about traditional tube smokers?
They can be used in professionally controlled low-temperature setups, but the difficult part is maintaining actual refrigeration conditions at the food, not generating smoke. Without a measured, controlled cold chamber and a validated food-safety process, I wouldn't recommend leaving cheese for hours in a garden barbecue, however fashionable the technique may be online.
Which wood?
Apple: mild and easy to enjoy. Maple: soft and rounded. Alder: delicate. Cherry: slightly more aromatic. Hickory can work in tiny amounts for robust cheddar, but it's easy to overdo. Start with one wood, a modest amount of smoke and cheese that actually tastes good without it.
What actually works on the plate.
Mature cheddar is the obvious choice because it already has enough bite to stand up to smoke. Gouda brings sweetness and a firmer texture. A firm alpine-style cheese gives you nuttiness that works brilliantly with an applewood note. Pick something you'd happily eat without the smoke. Smoking bad cheese just produces bad cheese with a new smell.
Take a taste before adding more smoke. Cheese is rich and holds aroma; what seems subtle at first can become assertive after the block has sat in the fridge. The best smoked cheese is the one that makes a sandwich better, not one that announces itself from the other side of the garden.
What to eat it with.
Use it in toasties with mustard, grate a little into mac and cheese, or serve with apples, pickles and crusty bread. Don't vacuum seal a homemade smoked cheese and claim that it will last for months because the outside smells like a smokehouse. Follow the original product's date and refrigeration instructions, use clean utensils and discard anything that has been left warm or handled unsafely.
It's an excellent little bit of kitchen theatre when done with restraint. But the point is still the cheese. If the smoke is the only thing you can taste, start again with less.