Building a flitch beam

[georgec]

I'm adding an addition to my one-story home. Home is 23ʼ 4" wide with center support. Addition is perpendicular to the long side of
the house, hence against the load-bearing wall. Addition is the same width as the house, 23ʼ 4". I plan to put a 14ʼ wide passage
between the existing house and the addition.

Roof and ceiling have support at the center of the house.

Existing roof deadload @ 15 psf so (15# * 11.67ʼ * 14ʼ ) /2 = 1225 load, or 87.5# per lf
New roof deadload on top of the existing roof is the same and must be supported = 1225 load, or 87.5# per lf
Existing ceiling @ 10 psf = 817 load, or 58# per lf
Snowload @ 60 psf ground, or 43 psf roof load = 3511 load, or 251# per lf
TOTAL = 6778 load or 484 per lf

I'm looking for a solution with minimum depth that fits in the existing 2x6 stud wall. Three LVLs don't handle it. I'm guessing that a
flitch plate is what I need.
Three 1 3/4" x 7" x 14 1/2ʼ LVLs + two pieces of steel, 7" x 14.5ʼ — But how thick?

Various different formulas give me 1/4, 3/8 and 1/2 for the thickness.

[ChugiakTinkerer]

Not sure about flitches, but I'd start with an I-beam or other structural steel.

https://webstructural.com/beam-designer.html