This is the subject of dispute. The Tur (OC 165) records that while Rashi's teacher (R Yaakov ben Yakar) ruled to wash twice and say Asher Yatzar in the middle, Rashi himself thought it was fine to wash once and say Asher Yatzar after finishing washing before saying HaMotzi. See there and in the Beit Yosef for how all the different Rishonim come out on the matter. The Rosh posits a compromise, whereby Asher Yatzar is recited after the first of the two pourings of water on the hands, where your hands are clean enough to say a Bracha, but are not ritually pure for bread. The Shulchan Arukh accepts this compromise position as valid if you don't want to wash completely twice. (Note in this case the "first" wash must be invalid in some way for bread, or else you're already done.)
The Mishna Berura there suggests that one can Bedieved (such as lack of sufficient water) rely on the "one wash" opinion and recite Asher Yatzar after Al Netilat Yadayim because of the many who hold that way because it is related to eating (as you said, like "feed the animals"). He does support "two washes" Lechatchila like the Shulchan Arukh.
The Arukh Hashulchan (ibid. :2) vouches for the "one wash" opinion even Lechatchila as many opinions hold that one must recite Al Netilat Yadayim when washing after the bathroom anyway (cf. OC 7).