A method is provided for manufacturing at least a portion of a heat exchanger. During this method, a first heat exchanger section is formed that includes a base and a plurality of protrusions. The forming of the first heat exchanger section includes building up at least one protrusion material on the base to form the protrusions. The first heat exchanger section is attached to a second heat exchanger section. A plurality of flow channels are defined between the first heat exchanger section and the second heat exchanger section.