Heat oven to 375°F.
Transfer goat cheese to a small bowl and place near stove top to warm slightly.
Dust a large cutting board with flour. Roll out your pastry sheet and sprinkle with a little more flour. Take a 10" skillet and flip it over onto the floured pastry. Using a knife, trace a line around the pan. Peal away the excess pastry and discard. You should be left with a 10" round of dough. Set aside.
Halve your shallots length-wise from root to tip. Peel back the skin and trim the dirty part of the root end. The root should still be in tact but look neat and pretty.
Heat your 10" skillet on medium-low. Add olive oil and arrange your shallot halves open-side down in the oil. Increase heat to medium and let cook until golden brown, about 5-10 minutes. Reduce heat and add 4 tbsp water. Cover and let cook for another 5-10 minutes, or until the shallots are soft and a knife can easily slide in and out without resistance.
Remove the cooked shallots and set aside on a plate. There might be a few shallots with dried and craggily outer layers. Peel those dry layers off and discard.
Back in the skillet, wipe away any excess water. Add balsamic vinegar and sugar, constantly mixing until the sugar has dissolved. Bring to a light simmer and let cook until slightly sticky, about 2 minutes. Add dijon and fresh thyme. Mix until combined (if there are still a few clumps of dijon, that's fine!). Finally, add cold butter and continue stirring until the butter has melted and a glossy, rich sauce forms.
Once the sauce is glossy, add your cooked shallots back to the skillet. Arrange them open-side down, so they each have a cozy little spot in the sauce.
Using a fork, prick several holes in the pastry round. Transfer the hole-y dough to the skillet to cover, and gently press down around the edges.
Transfer the skillet to the pre-heated oven and cook at 375°F for 40-45 minutes, or until golden brown and bubbly.
When the pastry is golden and flaky, remove from the oven. Let cool slightly (enough that you can touch the pan with your hand and not get burned, but hot enough that if you gripped the handle, and held onto it, it might burn you), then flip it over unto a cutting board. See the notes below for my tips on this!
If some of the shallots stick to the bottom of the pan, don't fret! Use a spatula, or spoon, to scoop them up and place them back on the tart. No one will ever know!
Cut the tarte tatin into 8 slices and top with a scoop of room temperature goat cheese and freshly sliced chives.
