To simplify the expression given, we need to handle both fractions and combine them.

The expression is:

\[
\frac{9 - 2\sqrt{3}}{3\sqrt{6} - 2\sqrt{2}} + \frac{3}{3 + \sqrt{6}}
\]

### Step 1: Simplify the first fraction

We can try to rationalize the denominator of the first fraction:

\[
3\sqrt{6} - 2\sqrt{2}
\]

To rationalize it, multiply the numerator and the denominator by the conjugate of the denominator:

\[
3\sqrt{6} + 2\sqrt{2}
\]

Now we have:

\[
\frac{(9 - 2\sqrt{3})(3\sqrt{6} + 2\sqrt{2})}{(3\sqrt{6} - 2\sqrt{2})(3\sqrt{6} + 2\sqrt{2})}
\]

Calculate the denominator:

\[
(3\sqrt{6})^2 - (2\sqrt{2})^2 = 54 - 8 = 46
\]

Now calculate the numerator using distribution:

\[
(9)(3\sqrt{6}) + (9)(2\sqrt{2}) - (2\sqrt{3})(3\sqrt{6}) - (2\sqrt{3})(2\sqrt{2})
\]

This results in:

1. \( 27\sqrt{6} \)
2. \( 18\sqrt{2} \)
3. \( -6\sqrt{18} = -6(3\sqrt{2}) = -18\sqrt{2} \)
4. \( -4\sqrt{6} \)

Combine the numerator:

\[
27\sqrt{6} - 4\sqrt{6} + 18\sqrt{2} - 18\sqrt{2} = 23\sqrt{6}
\]
So the first fraction simplifies to:

\[
\frac{23\sqrt{6}}{46} = \frac{23}{46}\sqrt{6} = \frac{1}{2}\sqrt{6}
\]

### Step 2: Simplify the second fraction

The second fraction is:

\[
\frac{3}{3 + \sqrt{6}}
\]

Rationalizing the denominator here gives us:

\[
\frac{3(3 - \sqrt{6})}{(3 + \sqrt{6})(3 - \sqrt{6})} = \frac{9 - 3\sqrt{6}}{9 - 6} = \frac{9 - 3\sqrt{6}}{3} = 3 - \sqrt{6}
\]

### Step 3: Combine both fractions

Now add both parts together:

\[
\frac{1}{2}\sqrt{6} + (3 - \sqrt{6})
\]

This results in:

\[
3 - \frac{1}{2}\sqrt{6}
\]

That’s the simplified expression. Thus, the final answer is:

\[
3 - \frac{1}{2}\sqrt{6}
\]