gramatická analýza: (Nejdřív jsem běžel) k té přejeté.

Past Passive Participle long form přejetý from přejet

The main verb (the conjugated verb) is jsem běžel; the subject of this sentence is therefore já. Being in the nominative position, já cannot be modified by přejetý which is governed by the preposition k which always takes the dative case. Looking at the ending -é in the dative, you can determine that přejeté cannot be other than fem. sg. dat (there is no dat. pl. form of an adjective ending -é). The PPP thus is modifying an implicit feminine noun žena, stará žena or stará žebračka (the woman/old female beggar who was run over (by the car)). Thus, the sentence could be paraphrased as "First of all I ran to that woman whom the car ran over."