The optimization process and recommendations:

1. Avoid Subqueries (query line: 12): We advise against using subqueries as they are not optimized well by the optimizer. Therefore, it's recommended to join a newly created temporary table that holds the data, which also includes the relevant search index.

The optimized query:

SELECT
        p.productid,
        p.productname,
        SUM(qty * od.unitprice) AS TotalAmount 
    FROM
        Production.Products AS p 
    JOIN
        Sales.OrderDetails AS od 
            ON p.productid = od.productid 
    WHERE
        N'USA' IN (
            SELECT
                o.shipcountry 
            FROM
                Sales.Orders AS o 
            JOIN
                Sales.OrderDetails AS od 
                    ON o.orderid = od.orderid 
            WHERE
                o.shipcountry = N'USA'
        ) 
    GROUP BY
        p.productid,
        p.productname