To refactor this...

//THIS IS PROBABLY BAD DESIGN, BUT I HAVE NO IDEA HOW TO IMPROVE.
    @Override
    public String cook() throws Exception {
        switch (recipe.getType()) {
            case BURGER:
                return cookBurger();
            case PIZZA:
                return cookPizza();
            default:
                throw new RecipeTypeNotExistException("Invalid recipe type: " + recipe.getType());
        }
    }

    // Method for cooking Burger 
    private String cookBurger(){...}

    // Method for cooking Pizza
    private String cookPizza(){...}

...enhance the business model with a new abstraction that represents cookers...

interface Cooker {

    Cooker EMPTY_COOKER = new Cooker() {
        boolean cooks(RecipeType type) { return true; }
        String cook(Recipe recipe) throws Exception { 
            throw new RecipeTypeNotExistException("Invalid recipe type: "
                                                 + recipe.getType());
        }
    };   

    boolean cooks(RecipeType type);
    String cook(Recipe recipe);
}

...with a corresponding implementation for each known cooker...

public class BurgerCooker implements Cooker {
    
    boolean cooks(RecipeType type) { return type == RecipeType.BURGER; }
    String cook(Recipe recipe) throws Exception { ... }
}

public class PizzaCooker implements Cooker {
    
    boolean cooks(RecipeType type) { return type == RecipeType.PIZZA; }
    String cook(Recipe recipe) throws Exception { ... }
}

...then enhance RecipeCooker class with a constant to keep in memory an instance of each available cooker and an instance variable of type Cooker that must be initialised by the constructor...

public class RecipeCooker {

    private static final Cooker[] COOKERS = { new BurgerCooker()
                                            , new PizzaCooker()
                                            };

    private final Recipe recipe;
    private final Cooker cooker;
    private final RecipeRepository recipeRepository;

    public RecipeCooker(Recipe recipe
                      , RecipeRepository recipeRepository) {
        this.recipe = recipe;
        this.recipeRepository = recipeRepository;
        Cooker toBeCooker = Cooker.EMPTY_COOKER;
        for ( int i = 0; i < RecipeCooker.COOKERS.length; i++ ) {
            if ( RecipeCooker.COOKERS[i].cooks(recipe.getType()) ) {
                toBeCooker = RecipeCooker.COOKERS[i];
            }
        }
        this.cooker = toBeCooker;
    }

    public String cook() throws Exception { 
        return this.cooker.cook();
    }
}

That would be the simples enhancement to keep consistent with the business model that includes a ReceipeCooker model.