What was the first TV show or movie where the viewer could hear a character's thoughts?

Wikipedia calls it Stream of Consciousness and it's a literary device dating back to the 19th century. More recently it's been referred to as Inner Monologue

The oldest movie might be Alfred Hitchcock's Murder! (1930), which has this section on TVTropes

Menier starts thinking about the case while he's shaving. In 1930, early sound recording technology did not allow for overdubbing—Herbert Marshall's dialogue could not be dubbed over the shot of him staring into the mirror. So Alfred Hitchcock had Marshall record the dialogue elsewhere, and the recording was played on the set as the camera rolled. This is believed to be the first-ever use of voiceover to convey a character's Inner Monologue. This might be the Trope Maker, as stage plays had the characters talk to the audience to reveal their thoughts.

This would explain why the trope is that such audio is tinny or distant

Any performance of Shakespeare

Shakespeare (and his contempories) used monologues extensively to show the audience the inner dialogue of his characters. Any broadcast of a performance of Shakespeare therefore qualifies.

Virtually all silent movies

In the absence of sound, silent movies naturally adopted the conventions of mime. Mime has very standard ways to show how a character is considering whether to do something. You see this many times in slapstick situations where the butt of the joke has their back turned, and the protagonist prepares to hit them with a stick/truncheon/whatever. Where the protagonist is definitely going to do it, they will gleefully wind up and prepare for the hit. And where the protagonist is unsure whether to do it or not, they will wind up, then turn away, then prepare again, then turn away again, and so on. The mental debate is clear, in spite of the lack of words.