"speaking" versus "speech"

Generally, there are three common usages of speech and speaking:

They are the following:

1. Speech (uncountable) is how some person sounds using language, or the ability to use language from sounds. It describes the physical features of the sound, such as tone and volume. It also describes a person's skill, style, or habits using language.

   Examples:

   • Humans are the only animal that uses speech.
   • Actors usually take speech training.
   • People of different countries often have very different speech.
   • Some health problems affect speech.

2. Speech (countable) is an event, usually planned, of one person talking to a group that listens only.

   Examples:

   • Some politicians become successful because of a single speech.
   • Martin Luther King's "I Have a Dream" speech is important in American history.

3. Speaking (uncountable, gerund of speak) is any specific act of communicating using speech.

   Examples:

   • We learned about problems in the community by speaking to locals.
   • I always stop playing music before speaking on the telephone.
   • Speaking at the same time as someone else is rude.

Also, because speaking is a gerund, it may have any usage of the simple form of the verb. For example, speaking clearly is important for my job is correct because it follows the same usage as I have to speak clearly for my job. Also correct, and with the same meaning, is clear speech is important for many jobs. However, humans are animals with speaking is not correct.

(In the case of the question, someone would make fun or your speech, because your speech is how you sound any time you talk, and not a single event or action.)

Speech is more normal in this context. Speaking is possible, but not idiomatic. However way of speaking is the most natural.

On another subject "Why do you making" is never grammatical in any context, as far as I can think. It mixes two different constructions:

Why do you make ...?

is the present simple;

Why are you making ... ?

is the so-called present continuous.

In practice, the "continuous" is much more common for most verbs, unless you are talking about something habitual.

So if you are talking about somethign going on at the moment,

Why are you making fun of my speech/way of speaking?

but if it's something that the person frequently does,

Why do you make fun of my speech/way of speaking?