SQL Server 2019

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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')/*,
(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	60	202007
12345	60	202008
12345	60	202009
12345	60	202010
12345	60	202011
12345	60	202012
12345	60	202101
12345	60	202102
12345	60	202103
12345	60	202104
12345	60	202105
12345	60	202106
12345	60	202107
12345	60	202108
12345	60	202109