You have a month and a day, so you can create a MonthDay (you will also need a year to create a LocalDate):
MonthDay md = MonthDay.parse(date, formatter);
If you want to use LocalDate, you can use MonthDay as a starting point:
int year = Year.now().getValue();
LocalDate localDate = md.atYear(year);
Or, alternatively, you can use the default year in your formatting:
DateTimeFormatter formatter = new DateTimeFormatterBuilder()
.appendPattern(pattern)
.parseDefaulting(ChronoField.YEAR, year)
.toFormatter(Locale.US);
LocalDate localDate = LocalDate.parse(date, formatter);
The advantage of this method is that it also checks for the correct day of the week (Thursday).