It is permitted to eat chicken after drinking milk even without cleaning out your mouth or eating something in between and washing your hands, but ideally you should have washed out your mouth, or drank something else.
Explanation:
Gemara Chulln 105a: Rav Chisda said one who ate meat is forbidden to eat cheese, one who ate cheese is permitted to eat meat. R’ Asi asked R’ Yochanan how long should you wait in-between, and he answered nothing.
Rashi explains since meat is very fatty and it gets stuck in your mouth and the taste stays for a long time unlike cheese. The Rashba (Torat haBayit pg. 103) explains that since cheese is soft it doesn’t get caught in the teeth.
Other Rishonim argue about why there is difference between meat and milk.
Rabbeinu Tam, and the Behag (Tosfot Chullin 104b) say that one who eats meat is forbidden to eat cheese, that is without washing and washing out you mouth but with washing your hands and mouth it is permitted. One who eats cheese can eat meat even without washing.
The Rif is of the opinion that after eating meat one need so wait six hours until he can eat dairy one who eats cheese needs to wash his hands and his mouth to eat meat and so rules the Rambam (Machalot Asurot 9:21).
In Shulchan Aruch (YOd 89:2) rules like the Rif, “one who eats cheese can eat meat immediately as long as you check your hands their is no residue from the cheese and at night where there is no light then you must wash your hands, and you must cleanout your mouth and rinse it by eating bread which cleans it well.”
The Rema writes some are stringent no to eat meat after cheese and the tis how we customarily do that by hard cheese we don’t even eat afterwards chicken. some are lenient and we should not protest as long as they do clean and rinse dn wash their hands. It is better to be stringent.
The Rema is stringent to wait six hours by hard cheese before meat like by cheese after meat. however, by soft cheese everyone agrees it permitted to eat it straight away, with cleaning and rinsing the mouth and washing.
Cleaning is with hard food. Rinsing is enough to drink and washing your hands from any residue.
Is there a difference between eating chees to drinking milk?
The Rashash (Chullin 103b) says that it is only milk which is liquid and it is enough to take a drink.
In Shu”t Divrei Malkiel understood the Rambam to hold you need to clean your mouth as well.
R’ Ovadia Yosef (Yechveh dat chelek 3:58) understood the Rambam like the Rashash as the Rambam writes (Maacholet asurot 9:26) he who ate cheese or milk it is permitted to eat meat after if he cleans his mouth and washes it before meat. “The Rambam leave out mil at the end of the halacha so it seems that only cheese needs washing.
R O” Y rules that one only needs to washout his moth after drinking milk before meat. Even the Gemara says,” Igra said chicken and cheese can be eaten however.” We see that which we are stringent with washing them only if you want to eat meat but chicken there is no reason to be stringent.
Yalkut Yosef also rules this way (Issur Vheter chelek 39:1) “One who eats cheese the basic halacha is that is permitted for hm to eat chicken afterwards without washing out the mouth we only wash by actual meat however he says you should be stringent (ben Ish Chai P Shelach pa14) even by chicken.
Even if you want to be stringent to wash between milk and chicken if you didn’t you did not transgress anything.
