After the take-home phase, each student comes in for a 15 minute oral exam. We go over each problem on the exam. If the work they've submitted looks correct, I'll ask the student to walk me through how they approached the problem. If it sounds like they understand the work they've submitted (and haven't just copied a classmate's work) I ask the student to define, explain or clarify a particular concept in the problem and connect it to the problem as a whole. If they can manage this, the real fun starts. I'll tweak one of the conditions in the problem and ask them how the change will affect their answer. I don't ask them to re-solve the problem on the spot typically; I just want to know if they really see all the dependencies and interrelations between the concepts.
I grade each problem on a 4 point scale with 4 being phenomenal work, and 0 indicating they couldn't even explain how they approached the problem. This also roughly corresponds to the A - F letter scale. Being able to talk your way through how you approached a problem gets you one point. Being able to clearly define, explain or clarify a chosen concept gets you another; this is C level understanding. The "tweak" usually occurs in two parts, an easier version and a harder version. Correctly answering gets you one point each. There are obviously shades of grey in all these, and I typically award partial credit in 1/3 point increments to make the +/- letter scaling work out well.
While I'm still working on the specifics, I love the take-home then oral exam model for a lot of reasons. I think it better simulates projects in the real world, both on time limits and collaboration. And with application-oriented students, connecting something to the business world gives you a ton of traction. One of my linear algebra students stated it really well (I'll paraphrase): "Taking a timed exam without resources or collaboration is like doing Mad Max mathematics." I couldn't agree more.
My understanding so far of the student experience is that they're initially really nervous about the idea of an oral exam, which is new to most of them. But they eventually really warm up to the format. Last semester, I put the format of the final exam up to an anonymous vote in class: should it be a traditional exam of a take-home+oral exam? Over 90% of students (N=60) voted for the nontraditional format. I don't think I could get 90% of them to agree on almost anything, so I took this as confirmation that something about this model was working for them.
For some examples, here's the midterm in applied calculus and the midterm in linear algebra. Notice in linear algebra I can use the midterm to preview things that we'll see more in depth later in the semester like multilinear and logistic regression. In calculus I can give them fairly gnarly