Prepare the Onion Compote: In a small saucepan over a medium-low heat, warm the olive oil. Add the thinly sliced onion, smoked paprika, caster sugar, and a generous pinch of salt. Cook gently, stirring occasionally, for 15-20 minutes, until the onions are deeply golden, soft, and almost jammy. The aroma will become sweet and rich. Once done, remove from the heat and keep warm.

Preheat the Grill: Fire up your gas or charcoal grill to a medium-high heat. You want a temperature of around 200°C (400°F). The grates should be clean and lightly oiled with a paper towel dipped in vegetable oil. You'll see a shimmering heat rising from the grates.

Grill the Sausages: Place the sausages on the hot grates, perpendicular to the bars. Grill for 4-5 minutes per side, rotating them a quarter turn halfway through if you want those classic cross-hatch marks. You are looking for a deep golden-brown colour and a slight charring. The skin will begin to blister and pop, and you'll hear a satisfying sizzle. The internal temperature should reach 71°C (160°F) for chicken or 74°C (165°F) for beef.

Toast the Buns: While the sausages are grilling, split your brioche buns and spread the cut sides with the softened butter. In the last minute of the sausages' cooking time, place the buns cut-side down on the cooler part of the grill. Toast for 30-60 seconds, until they are golden brown and have visible grill marks. The smell of toasted butter and bread is your cue.

Assemble with Care: Remove the sausages and buns from the grill. Place a sausage into each toasted bun. Spoon a generous amount of the warm onion compote over the top.

Add the Finishing Touches: Drizzle with American mustard and a spoonful of sweet pickle relish. Sprinkle with the finely grated cheddar cheese and a generous pinch of chopped fresh chives. The contrast of warm cheese against the hot sausage will be visually stunning.

Serve Immediately: Plate the hot dogs on a clean board or platter. The goal is to make each one look like an individual work of art. Serve straight away while the bun is still crisp and the cheese is just beginning to melt.