Is alcohol based sanitizer flammable?
Firstly, we should realise that sanitizers should not be considered as pure alcohols, alcohols are indeed a major component of it but many other important components are also present. A typical sanitizer contains these things -

Isopropyl alcohol/ Ethyl alcohol, Glycerin, Water, Carbomer(stabilizer) and some other components mostly related to cosmetics.

Now, i will answer your questions one by one

After applying sanitizers, can they catch fire?

Yes, there is 60%-70% alcohol content which is highly flammable. Organic compounds like alcohols, alkanes etc have very high enthalpy of combustion ($\Delta_\mathrm{c}H^\circ$ = -2000 kJ/mol for Isopropyl alcohol) and they can evaporate very easily due to the heat of our hands($\Delta_\mathrm{vap}H^\circ$= 40kJ/mol for isopropyl alcohol).

As soon as we apply alcohol, it evaporates so why is it still flammable?

This is actually wrong, as I earlier mentioned, that sanitizers are not only alcohols, in fact, the 'carbomer' present acts as a stabilizer to emulsify/dissolving all the components in it. These stabilizers prevent the alcohol to evaporate immediately and allow it to disinfect our hands fully. Also, after applying sanitizers, many of you might feel hands becoming sticky it is because of the presence of Glycerin and other cosmetic products used in it which act like a moisturizer. So generally it takes about 30-40 seconds for all(mostly) the alcohol to evaporate and sustaining a risk to be inflammable for a long time.
A thing to keep in mind is that the residue (mostly Glycerin) is also flammable(not that much, it's flammability rating is one) so after the alcohol has evaporated, you aren't fully safe from catching fire.
If your are working in kitchen, there is no actual need of using sanitizers as you can always wash hands in sink with soap and water.
I hope this answer cleared some of your doubts, stay safe :).