Which office permits your house
Inside the City of Los Angeles, the Department of Building and Safety (LADBS) reviews construction plans and approves the permit. Plans that need plan check go in online through ePlanLA. In unincorporated Los Angeles County and in the cities that contract with the County, the permit comes from the Public Works Building and Safety district office that serves the address.
If your lot sits in Beverly Hills or Malibu, the pages for those cities walk through the review there. Brentwood, Bel Air and Pacific Palisades sit inside the City of Los Angeles, with their own hillside and planning rules.
Which office your parcel answers to is the first thing I confirm. The full picture of who pulls the permit and what the inspector checks is on the permits and plan check page.