Extend Roof for Patio Cover
I agree with Ed Beal. But yes, there are 2 ways to do this &amp; they're equal in strength, for what's planning to be done. However, get a Permit from the Building Dept., so they can approve or correct the design, fasteners &amp; supporting post(s) dimensions &amp; footings. Plus, they can record that the patio was converted to a porch.

1st way is to do ledgers on both fasciae into structural members. Typically you'd pre-build the new structure so the short rafters are end screwed or nailed through the blue-line's ledger &amp; the long lower beam &amp; top ledger are through the side ledger (rafter). Also, use joist hanger type connectors to ensure nothing can pull away from each other.

2nd way is really the proper way &amp; is likely what the one contractor was talking about. You'd remove both fasciae &amp; screw or nail directly into the structure. For the slope side this is much less important. But, the blue-line top would have the new rafters cantilevered or sistered onto the home's existing rafters. This is much more secure &amp; even lets you re-use both fasciae.