Combine the plant-based milk and barley malt syrup, and heat in the microwave or on the stove to bring the temperature to 110°F. Any hotter than this and the yeast may die. Pour the mixture into the bowl of a stand-mixer, or if doing by hand, into a large bowl. Add in the yeast and let proof for 10 minutes.
Add in 2 cups of the flour, mixing well. Add in the oil and salt, and then the remaining 1 1/4 cups flour, adding more only as necessary. The dough should be very soft and a bit sticky. Resist the urge to add more flour at this point, less is more. Knead for 2 minutes, using a bit of oil on your hands and work surface if things are sticking, not flour.
Spray the bowl with oil, add the dough and then spray the top of the dough so it doesn't dry out. Cover with a light towel. Let rise for 1.5 hours. Punch down and then return the dough to the bowl, cover the bowl with cling wrap and place in the fridge overnight.
In the morning, remove the dough from the fridge. Preheat oil in a deep-fryer to 360°F.
Punch the dough down and knead into a smooth ball. Roll it out to 1/4" thickness on a very lightly greased work surface. If the dough is resisting, let it rest for 10 minutes (keep it covered under cling wrap). With a pizza cutter or a sharp knife, cut the dough into 2"x 2" squares.
Deep-fry, flipping every few seconds until the beignets become a golden color, about 2 minutes. Do not overcrowd the fryer. Keep the dough covered so it doesn't dry out while you fry in batches.
After the beignets are fried, drain them for a few seconds on paper towels. You can then move them to a wire rack to prevent sweating until cool, or to a serving plate and dust them with confectioner's sugar. If you'd like them completely coated, toss them in a bag with confectioner's sugar. Pinch the bag closed and shake to evenly coat the beignets. Serve immediately.