The verb start takes complement clauses with either the marked infinitive or -ing forms, and there is no consistent difference in meaning between the two.
Consequently, both of your examples are grammatical. However, the version with the infinitive will usually be better, as it avoids using two -ing forms close to each other. This preference is generally acknowledged in contemporary linguistics and goes by the name horror aequi principle, a Latin phrase meaning, literally “aversion to the same thing”:
The horror aequi principle involves the widespread (and presumably universal) tendency to avoid the repetition of identical and adjacent grammatical elements and structures. —Gunter Rohdenburg, “Cognitive complexity and horror aequi as factors determining the use of interrogative clause linkers in English”, in Rohdenburg and B. Mondorf, Determinants of Grammatical Variation in English, 2003.
Another article in the same volume, U.Vosberg, “The Role of extractions and horror aequi in the evolution of -ing complements in Modern English”, states that
In many languages, we observe a strong aversion to the (immediate) co-occurrence of identical or similar grammatical structures. Thus, the establishment of -ing complements has been significantly delayed after matrix verbs appearing in the shape of -ing forms themselves. Such contexts tend to favour either the to-infinitive as verb complement or other alternatives (like nominal objects or finite clauses).
As a rule, gerunds should be used like their verbal counterparts. If the verb is a phrasal verb, then the gerund should be constructed accordingly.
So your example "Thinking of you makes me happy" is correct, because you'd use "of" if you rephrased this with "think" as a verb:
When I think of you, it makes me happy.
However, this is incorrect
He has finished washing of the dishes.
because you'd say "I don't want to wash the dishes," not "I don't want to wash of the dishes." That last sentence makes no sense in English.
The Inevitable and Rarely Used Exception
You knew there was going to be an exception, right?
In a few cases, using the gerund + of is correct when it is important to emphasize the specific act the gerund describes. It also sounds extremely formal to an English speaker; it is not a construction one would use in everyday speech except as a joke.
Because this construction refers to a definite act, it must take the definite article "the." The phrase "She enjoys reading of poems" would sound odd to an English speaker, but "She enjoys the reading of poems" is perfectly correct, though having an elevated tone.
And because "the reading of poems" draws attention to the act and not the person performing the act, it can imply the reading of poems by others in addition to her own reading of poems.
A few more examples:
The signing of the Declaration of Independence actually took place on July 2.
The naming of new Nobel Prize laureates is always exciting.
I love the turning of the seasons.
Notice how the use of "the" refers to a specific act or event. The sentence
Signing the Declaration of Independence actually took place on July 2.
does not make sense because without "the," it implies a habitual action, not a specific act or event.
Humorous use of this construction usually makes an insignificant event sound more important than it really is:
Now is the time for the washing of dishes.
I humbly invite you to join me in the drinking of beer.
Best Answer
The correct structure is the one with to+infinitive:
In order to make the first example acceptable, it should be altered like this: