I work for exactly one of these product development consultancies / contract engineering firms - we are a team of ~10 EEs and SWEs who work with customers from idea to design to development and then manufacturing. To the point of "lots of subjects", the above includes but is not limited to: sales, project management, electronic schematic capture and PCB layout, design for and manufacturing support, web development, and mechanical/system integration. Often we'll collaborate with our mechanical engineering counterparts who've opened their own product development and contract engineering consultancy.

I can't speak for all firms but the way we find our employees is through networks of current employees and through LinkedIn recruiting. You'll want to emphasize aspects of your resume/work that mark you as a self-starting generalist with key skills.

An example on the generalist side: we had a candidate say, "I don't want to do design, I just want to be handed a schematic and do PCB layout." They were passed over in favor of another candidate willing to dive in and do schematic capture if necessary, PCB layout, and speak directly to customers about requirements, timelines, budgets, and other project planning logistics.

An example on the key skills side: the weakness of generalists is of course lack of depth, and we'll often contract out to SMEs based on project needs. If you know of a product consultancy near you that's, say, mechanically-focused, you can offer e.g. SWE expertise on, for example, frontend/backend development for a site that accepts RESTful HTTP requests from the IoT device they're building a mechanical enclosure for, whose customer needs software expertise that the MEs don't have.

To find these companies, useful search phrases are things like "product development", "contract engineering", "engineering consultancy" and so on.