Add flour, salt, water and olive oil to the bowl of a stand mixer.
Using the dough hook, knead the dough for five minutes. Add more water or more flour, if necessary. The dough should be homogeneous, not too sticky, not shaggy.
Transfer the dough to a well floured work surface and form it into a ball. Wrap in plastic wrap and allow the dough to rest for 30 minutes.
Set up a frying station. Line a baking sheet with parchment paper and a wire rack. Then, fill a heavy bottomed pot or dutch oven with olive oil about 3" deep and heat the olive oil to 350°F.
Cut off a chunk of dough, about the size of an egg, and cover the rest of the dough with plastic wrap so it doesn't dry out. Using a pasta machine or a rolling pin, roll the dough out to 1/8" in thickness.
Using a pastry cutter, or a knife, trim two sheets of dough so they are roughly 12 inches long. This will make three 4"x6" Fugassette once you are done assembling.
Space the cheese out on the dough so each Fugassette will be filled with one tablespoon of cheese. Leave a 1" margin around the edges, so that when you cut the Fugassette into rectangles, the cheese will be in the center. Sprinkle with dried thyme.
Place the other sheet of dough on top to cover the cheese and, using your fingers press around the cheese gently, eliminating as much air as possible.
Using a fluted ravioli cutter or a knife, cut the dough to make rectangles and crimp the edges with a fork.
Using a kitchen spider, carefully submerge the Fugassette in the hot oil and fry both sides until golden brown. Then, remove them from the oil and place them on the drying rack to drain the excess oil.
Fugassette are best when they are piping hot so keep them warm in the oven until you are ready to serve. Enjoy!