Prepare the squash by first cutting off the very top and stem. Then, separate out a solid cylinder of squash by cutting just above where the seeds start and keeping the middle piece. Now, peel the skin off that center portion and cut it in half, top to bottom. You will end up with two semi circle pieces. (At this point, I recommend weighing the prepped squash, so you know if there’s a need for a second squash.) From there, use a mandolin or slicer attachment on a food processor to cut .08in (2mm) thick pieces of squash. Set the slices aside.
Now, prepare the filo (phyllo) by taking out a single sleeve from the box and unrolling it, so the layers are all laying flat. Place them on a baking tray with cling film and then a towel over the top of them. This will keep the filo from drying out while you shape the soufra.
When the filo is prepped, make the sage infused-browned butter. First, chop up the sage into small pieces or ribbons. Then, heat the butter in a small saucepan on medium heat. Make sure to continuously stir or swirl the pan as the butter melts and then froths. Add the sage to the pan when it starts to froth. Keep stirring and swirling. When you see the milk solids at the bottom of the pan turn a light brown color, turn off the stove and move the pan from the heat. The butter will continue to cook and should end up a nice warm brown.
Preheat the oven to 350℉ (177℃). Then, move the browned butter pan, the sliced squash, and the filo pan to a counter top with a good amount of free space. Butter, oil, or spray the baking dish with cooking oil to grease it and place that in the workspace as well. To start, take a single sheet of filo and lay it flat on the counter. Then, use a pastry brush to gently spread the browned butter until the sheet is coasted end to end. Press a second piece of filo on top of the first and smooth them out.
Next, take two slices of squash and place them at the top of the filo sheets with some of the squash sticking over the edge. Accordion the filo so it folds up, over, and back down. The first row of squash should be in a kind of pocket, still protruding from the filo. Make the next row by laying a single piece of squash on the filo in the same manner as the first two. Fold the filo again, so the single slice is sticking out just as much. Next, you guessed it, use two more slices and make the same accordion fold. Continue adding slices in alternating number (1 or 2) and fold until you have used the first set of filo sheets. Place the filo and squash segment into the greased dish and begin a new one. (If the filo section ends a little too high but isn’t long enough to make another row, just fold it over to match the other fold heights.)
Make another segment. Butter the next filo sheet and top it with a second sheet. Then repeat the accordion folds with alternating amounts of squash slices. Place the second segment and keep repeating this process until only one filo sheet is left. This sheet can get tossed or buttered. If you butter the last sheet, fold it over itself. Then, add squash and fold until it’s used up. Add the smaller segment to the dish and baste the whole thing with any remaining browned butter. when basted, put the baking dish into the oven for 25 minutes.
While the soufra is going through it’s first bake, make the custard. Grate the cheddar cheese and chop the rest of the sage, then set them aside. In a small pot, add the cream and turn the heat to medium/medium low. Heat the cream to 150℉ (65.5℃) and keep it within a few degrees of that temperature. Then, add in the salt, pepper, sage, and nutmeg. Stir in the cheese and spices until the cheese has melted all the way and the sauce thickens but stays creamy. Take the pot off the heat and set it aside.
In a medium sized bowl, crack the eggs and whisk them until they froth up. They should be so mixed that you cannot differentiate between the egg whites and egg yolks. At this point, check the temperature of the cheese sauce. It must be below 140℉ (60℃) before it’s added to the eggs or they will scramble. When the sauce is in the correct temperature range, temper the sauce into the eggs by slowly pouring and constantly whisking it into the eggs. Once all of the cream is added, set the custard aside.
After 25 minutes, remove the squash and filo from the oven and let it cool for just a few minutes on the counter. Do not turn off the oven. When the filo has cooled enough to touch, pour the custard over each row. This works best if you can get custard into each pocket as far down as possible.
Return the cheesy squash and sage soufra to the oven and bake for another 25 minutes. When done, remove the soufra and let it cool for another 5-10 minutes. At this point, the soufra is ready to cut into portions and serve.