The optimization process and recommendations:

1. Avoid Subqueries (query line: 5): 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.
2. 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.

Optimal indexes for this query:

CREATE INDEX employee_idx_employee_id ON employee (employee_id);
CREATE INDEX employees_teams_idx_team_code ON employees_teams (team_code);

The optimized query:

SELECT
        AVG(e.days_of_employment) avg_days_of_employment 
    FROM
        employee e,
        (SELECT
            DISTINCT employees_teams.employee_id 
        FROM
            employees_teams 
        WHERE
            employees_teams.team_code IN (
                'ERA', 'RR'
            )) available_employees 
    WHERE
        e.employee_id = available_employees.employee_id