Here you go. Leave the computing to PHP and save your DB some work. This way you can make effective use of an index on the Date
column.
<?php
$date = $_POST['period'];
$start = strtotime($date);
$end = strtotime($date . ' 1 month - 1 second');
$query = sprintf(
'SELECT *
FROM projects
WHERE Date BETWEEN FROM_UNIXTIME(%u) AND FROM_UNIXTIME(%u)',
$start,
$end
);
EDIT
Forgot the Unix timestamp conversion.