The chicken and pesto panini is defined by a tension the press creates: a dry protein and an oily sauce being heated and compressed together. Cooked chicken breast is lean and goes stringy and arid under heat with nothing to carry it; basil pesto is loose, oil-heavy, and prone to running once it is warm. Put them in a roll and clamp it between hot ridged plates and the build either resolves into something where the oil rescues the chicken, or it splits, with pesto pressed out the open sides and dry meat left inside. The defining decision is how those two are arranged so the press makes them one thing rather than driving them apart.
The craft is moisture routing and load control. The pesto is used as the lubricant the chicken does not have, spread against the meat so its oil bastes the breast as the panini heats rather than pooling on the crumb and being squeezed out. The chicken goes in sliced or pulled rather than as a thick block, so heat reaches the centre before the bread over-crisps and the slices compress flat instead of rolling under the plates. Often a melting cheese, mozzarella or a mild Cheddar, is added not as a third flavour so much as a binder: it melts into the gaps and glues the dry meat, the oily sauce, and the bread into a single fused layer that holds when cut. The roll is a close, firm-crumbed one that takes the clamp and crisps, and it is filled and pressed to order, because a chicken-and-pesto panini left to stand goes from molten to leathery and the oil sets greasy.
The variations are mostly what gets added to manage the dryness or push the flavour. Sun-dried tomato or roasted red pepper brings sweetness and a little moisture against the lean meat; rocket added after pressing keeps a fresh, peppery counter that heat would wilt. A mozzarella-heavy build leans on the melt; a red-pesto version trades basil for a tomato-based sauce with the same oily function. The plain mozzarella and tomato panini solves the opposite problem, too much water rather than too little. Each deserves its own article rather than being crowded in here.