Overcome SubQuery Returned More Than 1 Result
No GROUP BY, no subqueries needed. The answer assumes that @Two has only one row per employee:

SELECT 
    a.employee,
    [EOD] = a.eod,
    [Paid] = CASE WHEN a.eod = MIN(a.eod) OVER (PARTITION BY a.employee)
                 THEN b.amtpaid
                 ELSE 0
             END
FROM 
    @One a
    JOIN @Two b
    ON  a.employee = b.employee ;


Another way would be to use window functions and a left join:

SELECT 
    a.employee,
    [EOD] = a.eod,
    [Paid] = COALESCE(b.amtpaid, 0)
FROM 
    ( SELECT a.employee, a.eod,
             rn = ROW_NUMBER() OVER (PARTITION BY a.employee
                                     ORDER BY a.eod)
      FROM @One AS a
    ) AS a
    LEFT JOIN @Two b
    ON  a.employee = b.employee 
    AND a.rn = 1 ;


which would be easier to modify if @Two has more than one rows per employee and we want the sum:

SELECT 
    a.employee,
    [EOD] = a.eod,
    [Paid] = COALESCE(b.amtpaid, 0)
FROM 
    ( SELECT a.employee, a.eod,
             rn = ROW_NUMBER() OVER (PARTITION BY a.employee
                                     ORDER BY a.eod)
      FROM @One AS a
    ) AS a
    OUTER APPLY
    ( SELECT amtpaid = SUM(b.amtpaid)
      FROM @Two b
      WHERE a.employee = b.employee 
        AND a.rn = 1
    ) AS b ;