Preheat oven to 375°F; have ungreased shiny cookie sheets ready.
In a large mixing bowl, with electric mixer, cream together the butter and sugars; next, beat in egg, peanut butter and vanilla.
In a smaller bowl, sift together salt, baking soda and flour (I do this through a sieve, not with a sifter).
With a wooden spoon, stir flour mixture into peanut butter mixture until a dough is formed.
If desired, you could stir in about 1/2 cup (or more) of chocolate chips now; I usually don't (even though I'm a chocoholic) as I often just want a pure peanut butter cookie with nothing else in it, not even peanut bits.
Stash dough in fridge while you clean up the dishes; this step isn't absolutely necessary but the fridge slightly firms up the dough, making it easier to work with.
Drop by heaping teaspoons onto cookie sheets and press down, criss-cross style, with a fork-- the classic peanut butter cookie look.
If fork sticks to cookies too much as you press down, lightly flour the fork.
Bake in preheated oven for 7 to 9 minutes, or until edges are slightly darkened (I like to cook these at 7-1/2 minutes).
Do not worry if they don't look done; cookies firm up as they cool and if you want chewier cookies it is very important not to overbake them.
Add an extra minute if you like crispier cookies.
Remove cookies from sheets after letting them rest on sheets for one minute, then let cooked cookies cool on racks.
Store cookies in a tin or in cookie jar (airtight containers are best).
I use a one-inch scoop to make the cookies; the approximate yield I get is 4-1/2 dozen cookies.