I was wondering how I could make a table similar.
You can't really, and shouldn't try. Making a table that looks similar is easy enough, but actually laminating solid wood to plywood is inherently a flawed methodology. Even when you do this with veneer it can cause bowing if it's not done correctly, and veneer is extremely thin, so it's easy to imagine the problems that might arise with a much thicker wood bonded to the plywood.
That is not the only issue with this design if we can see the details of the construction correctly from the pictures (the other major one shows up at the edges). As a result I very much doubt examples of the Melissa Table won't see structural problems down the line, it seems a certainty.
Off-topic but I also think the "reclaimed old-growth pine" thing either describes just some of the wood, perhaps individual tables in the series, but possibly none of them at all... unless their definition of 'old-growth' is wood from tract houses from the '60s or '70s! Look at the short lengths, the abundance of knots and most importantly, the wide growth rings in the example pics on the site. The lack of narrow growth rings alone is suspicious, but taken all together you can be certain someone is fibbing or at least stretching the truth :-/
It looks like in this table they used veneer or really thin wood to cover the edges of the planks and plywood.
That does look like how they did the edges but even at that it's not sound construction, the short sides are the particular concern.
So how to build a table that does look somewhat like this, but do it stably? The three best ways I can think of are:
- use solid wood of the full thickness (thicker wood is more costly but in pine this wouldn't be exorbitant);
- laminate thinner boards to the final thickness (there shouldn't be issues with movement as each layer will have approximately equal response to changes in moisture level);
- use plywood that you veneer with shop-made veneer (and importantly, counter-veneer also), then do the edges just as this table appears to have done it.