The optimization process and recommendations:

1. Create Optimal Indexes (modified query below): The recommended indexes are an integral part of this optimization effort and should be created before testing the execution duration of the optimized query.
2. Prefer Inner Join Over Left Join (modified query below): We identified that one or more left joined entities (e.g. `tdb_sales`) are used in the 'where' clause, in a way that allows to replace it with an optimized inner join. Inner joins can be fully optimized by the database, while Left joins apply limitations on the database's optimizer.

Optimal indexes for this query:

ALTER TABLE `tdb_products` ADD INDEX `tdb_products_idx_prod_id_prod_name` (`prod_mp_id`,`prod_name`);
ALTER TABLE `tdb_sales` ADD INDEX `tdb_sales_idx_sl_day_sl_mon_sl_year` (`sl_day`,`sl_mon`,`sl_year`);
ALTER TABLE `tdb_sales_temp` ADD INDEX `tdb_temp_idx_vt_cupom` (`vt_cupom`);

The optimized query:

SELECT
        DISTINCT p.prod_name,
        p.prod_price,
        Sum(dt.vt_qtd) AS total_qtd 
    FROM
        tdb_products p 
    LEFT JOIN
        tdb_sales_temp dt 
            ON p.prod_mp_id = dt.vt_product 
    INNER JOIN
        tdb_sales s 
            ON dt.vt_cupom = s.sl_coupom 
    WHERE
        s.sl_day = $day_link 
        AND s.sl_mon = $mon_link 
        AND s.sl_year = $year_link 
    GROUP BY
        p.prod_name 
    ORDER BY
        p.prod_name ASC