Preheat the oven to 190C/375F/Gas 5.
To make the filling, mix the butter with the garlic, parsley, lemon zest, lemon juice and a pinch of salt and pepper, to taste, in a bowl. Shape the flavoured butter into two logs, using cling film to help you wrap and roll. Chill until firm.
Make a pocket in each chicken breast by cutting a slash lengthways through the breast. Cut the butter logs in half and place one half in each chicken pocket. Secure with cocktail sticks.
Mix the flour, paprika and some salt and pepper together in a shallow bowl. Tip the beaten eggs into another shallow bowl and the breadcrumbs into a third shallow bowl. Dredge the stuffed chicken breasts in the flour to coat, then into the egg, turning until covered. Repeat the flour and egg coating once more, then dip into the breadcrumbs and coat completely.
Pour the oil into a frying pan to a depth of about 1cm/½in and set over a high heat. Check that the oil is hot enough, and then lower the chicken breasts carefully into the pan. Spoon hot oil over the top to seal the crumbs for about a minute. Turn over to brown the other side, again spooning oil over the top until golden-brown; this should take a further 1-2 minutes.
Remove the chicken from the oil and place in a shallow roasting tin. Bake the chicken in the oven for 10-12 minutes, or until the breasts feel firm when pressed. Remove and drain on kitchen paper. Carefully remove the cocktail sticks and serve.