It’s not the person who does the process, it’s the company. How do you propose (the company) filing for an employment based green card if you don’t even have a work visa yet ?? Your company lawyer should know this, even if they just google it. Plenty sites explain the process. By the way, neither the H1b nor any potential green card application thereon are automatically approved. In context with the other thread, the company you are thinking of clearly knows zero about this process. Why are they so keen to do something complicated and expensive they know nothing about for a new graduate? Do you have a relative working there?
H1B is a nonimmigrant visa category. You can start a green card process without H1B, though based on the following employment-based preference category:
The Employment First Preference EB1 category is for “priority workers.” These include persons with “extraordinary” ability, outstanding professors and researchers, and managers or executives of multinational corporations.
The Employment Second Preference EB2 category includes persons with “exceptional” ability and professionals with an advanced educational degree or equivalent experience.
The Employment Third Preference EB3 category includes certain professionals and skilled or unskilled workers.
The Employment Fourth Preference EB4 category is for certain special immigrants. There are many subgroups within this category, covering highly specific professions and circumstances.
The Employment Fifth Preference EB5 category is for immigrant investors. To be eligible in this category, a prospective immigrant must invest, without borrowing, a very substantial amount of money in a job-creating commercial enterprise in the United States.
H1 is dual intent. Perhaps George missed the other thread where the OP was enquiring about a job offer from a company that included H1 ( but that this company for some odd reason required OP to first do all the research on his/her own about the process before hiring a lawyer) which was what I specifically based my answer on. Certainly there are other options, should the OP meet the criteria. An experience immigration attorney is probably highly desirable in all these cases, considering both OP and his company appear to have no experience in employment based visas or green cards.