So there will be one arm, one foot long, that is supporting the sign? I assume this is inside the house? 150 pounds is pretty heavy, so here’s how I would do it:
Face plate, 2x6 by depth of sign (at least), which will be secured to the stud. Deck screws as beowulf suggests, two across every six inches should be enough. If you want no extra width on the face plate after the braces are installed, rip it down to 4-1/2 inches wide.
Before securing face plate to wall, attach support arm 2x4 centered at top of face place from the back of the face plate; no more than 3 deck screws strategically placed, more will weaken the support arm. The profile of the support arm will be vertical rather than horizontal; there will be 2 inches on either side of the support arm on the face plate (or 1-1/2 inches if you ripped down the face plate).
Make two 2x4 braces with angle cuts to go on either side of the support arm. The top angle will end up being horizontal, the side angle will end up being vertical; the two angles together should add up to 90 degrees. Attach the two braces at the top with screws to the support arm (or bolt through all three pieces); attach the two braces at the bottom to the face plate (you can screw from the back or the front, depends on how you want it to look; watch out for excess screw length if you screw from the back). Make sure the braces are plumb (i.e. vertically straight from the front).
Screw the whole assembly to the wall as in the second paragraph. Then, if you weigh more than 150 pounds, test it by hanging on it yourself, and bouncing a little. If it seems solid, hang your sign. If it seems to have any give at all, replace screws with lag bolts (and countersink them for a better appearance).
You can dress this up in a variety of ways to look good, paint being the easiest. If you’re ambitious you can countersink all the screws or bolts that show and put in wood plugs (which you can buy, or make with a plug cutter). If you pick a lower grade of lumber with knots etc. you can spackle those smooth before painting.
Voilá.
Roddy
eta: posted before I saw your second post. The method I described would be if you weren’t using metal brackets. If you do use metal brackets, I would still recommend testing it by hanging from it yourself. I wouldn’t be worried about the metal brackets coming out of the wall, but the support arm coming off the bracket. Not sure what size support arm you are proposing, you may need 4x4 instead of 2x4. That would still fit on the stud.