Region of Operation of MOSFET and current cutoff frequency
Disclaimer: the answer below relates to a mosfet transistor used as a voltage amplifier.

Summary
Transition frequency \$f_T\$ is a figure of merit (FOM) of a transistor reflecting both frequency response and gain. The value of \$f_T\$ does not represent unity gain frequency of a transistor, thus, the name “transition frequency” is indeed misleading, imho. Unity gain frequency of a transistor amplifier is driven by input and output RC constants largely (not transition frequency \$f_T\$). The value of mosfet’s gate oxide capacitance is a major factor driving frequency response and the value of transconductance \$g_m\$ governs the voltage gain.

Details
1)  “The current cut-off frequency for a MOSFET is defined as the frequency at which current gain (ratio of drain to source and gate to source current) is unity.”
This is called transition frequency. See here or [1] below, page 291. 3dB frequency (aka cut-off frequency you mentioned) is a different term. Please, amend the wording, if agree.  

2)  To see what drives frequency response of a transistor, let’s look at the following example, conventional common source NMOS amplifier:







simulate this circuit &ndash; Schematic created using CircuitLab

This amplifier has the following transfer function:
$${A_V}(f) \equiv {{{v_{out}}} \over {{v_{in}}}} =  - {A_{V,LF}} \times {{(1 - j{f \over {{f_z}}})} \over {(1 + j{f \over {{f_{p1}}}})(1 + j{f \over {{f_{p2}}}})}},$$
where \${A_{V,LF}} = {g_m}{R_{load}}\$ is a low frequency gain, \${f_{p1}} = {1 \over {2\pi {R_s}{C_{in}}}}\$ and \${f_{p2}} = {1 \over {2\pi {R_{load}}{C_{out}}}}\$ are poles governing frequency response, and \${f_z} = {{{g_m}} \over {2\pi {C_{gd}}}}\$ is a right-half plane zero,

$$\eqalign{
  &amp; {C_{in}} = {C_{gs}} + {C_{MI}} = {C_{gs}} + {C_{gd}}(1 + {A_{V,LF}}) \approx {C_{gs}} \approx {2 \over 3}{C_{ox}}  \cr 
  &amp; {C_{out}} = {C_{MO}} = {C_{gd}}(1 + {1 \over {{A_{V,LF}}}}) \approx {C_{gd}} \cr}. $$

It was assumed that output resistance of the mosfet is much greater than \$R_l\$. Note also that \${C_{gd}} &lt;  &lt; {C_{gs}}\$.

See [1] pages 661-662 for derivation of the formulas.

So, the first pole \$f_{p1}\$ is governed by gate to source capacitance and source resistance (and to a lesser extent by gain through input miller capacitance assuming \${C_{gd}} &lt;  &lt; {C_{gs}}\$, what is usually the case for mosfet in saturation mode).

The second pole \$f_{p2}\$ is governed by value of load capacitance (in this example Miller output capacitance of gate to drain capacitance \$C_{gd}\$) and load resistance.

Now, we see that gate oxide capacitance \$C_{ox}\$(or gate to source capacitance \$C_{gs}\$, which is around 2/3 of the total gate oxide capacitance, whichever you like) is the major factor driving frequency response. The major factor driving low-frequency gain is transconductance \$g_m\$.

3)  In practice we want both high cut-off frequency and high gain. Thus, we want high transconductance and low input capacitance simultaneously. The parameter \${{{g_m}} \over {{C_{ox}}}}\$ would be a proper FOM. Since \$f_t\$ is proportional to \${{{g_m}} \over {{C_{ox}}}}\$ (see below) it became such a FOM. 

4)  What is \$f_t\$ and how to boost it?
\$f_t\$ is defined as a short-circuit (small signal gain, strictly speaking) current gain of transistor. 

It can be derived that \${f_T} \approx {{{g_m}} \over {2\pi {C_{gs}}}}\$ (see the answer of Vineesh or [1] page 290).
Applying long-channel formula for transconductance it is turned into:
$${f_T} \approx {{{g_m}} \over {2\pi {C_{gs}}}} = {3 \over {4\pi }}{{\mu {V_O}} \over {{L^2}}} \propto {{\mu {V_O}} \over {{L^2}}},$$ where \${V_O} \equiv {v_{GS}} - {V_{TH}}\$ is overdirve voltage and \$\mu \$ is charge carriers mobility.   \$g_m\$ is proportional to \$C_{ox}\$. \$C_{gs}\$ is also proportional to \$C_{ox}\$, thus \$C_{gs}\$ has disappeared (was crossed out).

So, to increase transition frequency:
1) use semiconductor with higher charge carriers mobility (hard to implement). From this, we see that nMOS has higher \$f_t\$ than pMOS due to higher electrons mobility than holes;
2) use higher overdrive voltage to boost \$g_m\$, the drawback is smaller output range;
3) use minimum gate length (the drawback is lower output resistance which may deteriorate gain).

[1]: “CMOS Circuit Design, Layout, and Simulation, 3rd Edition”, R. Jacob Baker