The Croque-Bolognaise is a French sandwich that has borrowed an Italian sauce and not entirely worked out what to do with it. The structure is the Croque-Monsieur base of pain de mie, cheese, and béchamel, with a spoonful of meat ragù layered into the middle in place of the ham. Sometimes the ham stays and the ragù sits on top of it; sometimes the ragù replaces the ham outright. The cheese stays Gruyère or Emmental, the broiler step stays, the result is somewhere between a croque and a small open lasagne.
The version works best when the ragù has been cooked down properly. A wet sauce, freshly stirred and still loose with tomato juice, will defeat the bread within a few minutes of broiling and leave the sandwich structurally compromised. A ragù that has been simmered until it is nearly dry, with the tomato reduced and the fat from the beef and pork sitting visibly on top, behaves more like a paste between the slices and lets the bread keep its integrity. The cheese on top has to be generous enough to seal the ragù in and brown into a proper lacquer, otherwise the heat of the broiler dries the meat out before the bread crisps. A few cooks use mozzarella instead of Gruyère in this version to push the Italian register, though the canonical French preparation stays with the Alpine melt-cheese on the grounds that the sandwich is still a croque, not a pizza on bread.
The variant sits inside the cluster of croques that borrow from neighbouring cuisines: the Croque-Hawaiien with pineapple (which the French sandwich shelf tolerates more easily than the Italian sandwich shelf would), the Croque-Mexicain with chilli and avocado, the Croque-Tartiflette that crosses the line into a casserole. All of these belong to the broader Croque-Monsieur tradition, which has proven remarkably elastic about what it will accept as a filling so long as the bread, the melted cheese, and the broiled top are in place. The Bolognaise version is most often found at brasserie lunch menus and at chain cafés that serve a casual cross-section of comfort foods, and is rarely encountered in the more traditional cafés where the canonical ham version remains the default.