Combine the lukewarm water, yeast, and inulin in a small bowl with one gentle stir of a fork, then cover the bowl with foil and leave it aside. The mixture should look moistened throughout, without dry pockets of powder.

Distribute the vital wheat gluten, allulose, oat fiber, lupin flour, coconut flour, baking powder, xanthan gum, and salt evenly with a whisk in a separate bowl. A uniform dry mixture helps avoid concentrated patches of gum or leavener.

Work the cubed butter through the dry mixture with a fork or pastry blender until the pieces resemble peas, then pour in the cold water, cream, and reserved yeast mixture. Stop cutting while individual butter pieces remain clearly visible.

Bring the mixture together with a spoon, gather it into a ball, and enclose it in plastic wrap before refrigerating for 30 minutes. The dough should hold together but still show pieces of butter.

Set the chilled dough on a work surface lightly dusted with vital wheat gluten and lay parchment over it before rolling. The parchment creates a barrier between the dough and rolling pin without heavy dusting.

Lengthen the dough into a rectangle, fold the two short ends inward in overlapping thirds, neaten the edges, rotate 90 degrees, lightly dust the surface and dough again, and continue until you have completed four folds. If rolling becomes difficult or the dough springs back strongly, wrap it and refrigerate for 30 minutes before continuing.

Refrigerate the wrapped dough for 30 minutes, then give it two additional rolling-and-folding turns, stopping when the butter appears mostly incorporated as streaks rather than chunks. Use another 30-minute refrigerator rest if resistance returns, and look for a marbled appearance rather than a greasy paste.

Wrap the folded dough and refrigerate it for at least 30 minutes before dividing. The chilled dough should feel firm enough to handle without smearing the butter.

Unwrap the dough, divide it into equal halves, and return one half to the refrigerator in fresh wrapping. Keeping the unused portion chilled helps preserve its butter layers while you shape the first portion.

Roll the exposed half in both directions to make a broad rectangle, taking it as thin as the dough comfortably allows. Aim for even thickness rather than forcing tight dough into a larger sheet.

Divide the rectangle into three smaller rectangles, cut each diagonally into two long triangles, then stack each pair and roll it into one elongated triangle. Each pair forms a single croissant, giving you three triangles to shape from this half.

Coil one triangle from its broad cut edge toward its point, then press the point gently into the rolled dough. The finished roll should hold its shape with the tip secured against the surface.

Shape the remaining two paired triangles in the same way. Keep the rolls similar in size so they bake evenly.

Arrange the shaped pastries with space between them on a parchment-lined baking sheet, then retrieve the refrigerated half and repeat the rolling, cutting, pairing, and shaping. You should now have six croissants with room to expand.

Beat the egg until uniform, brush a light coating over the pastries, cover them with plastic wrap, and let them rise for 1 hour at 75°F to 80°F, away from a hot spot. Look for slightly puffed shapes with defined edges rather than greasy surfaces or pools of melted butter.

Start preheating the oven to 375°F (190°C) during the final 20 minutes of rising, remove the plastic wrap, and bake for about 20 to 25 minutes, adding a loose foil covering if the tops darken too quickly. The finished croissants should have golden-brown ridges rather than scorched tops.