SQL Server 2019

By using db<>fiddle, you agree to license everything you submit by Creative Commons CC0.

CREATE TABLE dbo.GLProject
(project int, amount int, period date);

INSERT dbo.GLProject VALUES
(12345,	10, '20200101'),
(12345,	10, '20200201'),
(12345,	10, '20200301'),
(12345,	10, '20200401'),
(12345,	10, '20200501'),
(12345,	10, '20200601'),
(12345, 10, '20200901')/*,
(23456, 20, '20210901')*/;

DECLARE @d date = DATEFROMPARTS(YEAR(getdate()), MONTH(getdate()), 1);

;WITH src AS -- mostly your existing query
(
  SELECT project, 
         period,
         rn = ROW_NUMBER() OVER 
              (PARTITION BY project ORDER BY period DESC),
         amount = SUM(SUM(amount)) OVER 
                  (PARTITION BY project ORDER BY period
                   ROWS UNBOUNDED PRECEDING)
  FROM dbo.GLProject
  GROUP BY project, period
),
recur AS
(
  SELECT project, period = DATEADD(MONTH, 1, period), amount
  FROM src 
  WHERE rn = 1 AND period < @d
  UNION ALL
  SELECT project, DATEADD(MONTH, 1, period), amount
  FROM recur WHERE period < @d 
)
SELECT project, amount, period = CONVERT(char(6), period, 112)
FROM src
UNION ALL
SELECT project, amount, period = CONVERT(char(6), period, 112)
FROM recur
ORDER BY project, period;

project	amount	period
12345	10	202001
12345	20	202002
12345	30	202003
12345	40	202004
12345	50	202005
12345	60	202006
12345	70	202009
12345	70	202010
12345	70	202011
12345	70	202012
12345	70	202101
12345	70	202102
12345	70	202103
12345	70	202104
12345	70	202105
12345	70	202106
12345	70	202107
12345	70	202108
12345	70	202109
12345	70	202110
12345	70	202111
12345	70	202112
12345	70	202201
12345	70	202202
12345	70	202203
12345	70	202204
12345	70	202205
12345	70	202206
12345	70	202207
12345	70	202208
12345	70	202209
12345	70	202210
12345	70	202211
12345	70	202212
12345	70	202301
12345	70	202302
12345	70	202303
12345	70	202304
12345	70	202305
12345	70	202306
12345	70	202307
12345	70	202308
12345	70	202309
12345	70	202310
12345	70	202311
12345	70	202312
12345	70	202401
12345	70	202402
12345	70	202403
12345	70	202404