Getting peanut butter out with the back of a spoon

The verb for this, which doesn't specify the part of the utensil used, is actually spoon:

[Merriam-Webster]
transitive
: to take up and usually transfer in a spoon

Used in your example:

I spooned peanut butter out of the jar.

What's good about this verb is that it doesn't matter how you used the spoon, only that you did.

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However, if you do want to specify the exact method, you could simply add the description.

• Assuming you don't mind the repetitive use of the word:
  I spooned out peanut butter with the back of a spoon.
• If you don't like the repetition, you can do what I did at the start of this answer:
  I spooned out peanut butter with back of the utensil.

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To address a comment that the Merriam-Webster definition cannot be interpreted as functional with the back of a spoon, because it uses the preposition in, here is the Oxford (Lexico) definition of the verb:

1 [with object and adverbial of direction] Put (food) into or on something with a spoon.
‘Rosie spooned sugar into her mug’

Note how it uses the preposition with rather than in.

As such, I don't hold to the inference that spoon necessitates a specific usage.