Is it this recipe? I see that she uses bread flour, and just says knead until smooth. How are you working the dough - by hand, in a food processor, in a stand mixer? My guess would be that they may seem tough because you are working in too much extra flour in kneading and rolling onto floured surfaces. If it's this recipe (which is quite a large volume, 7 cups of flour?) they give the oil temperature at 350F.
http://www.foodnetwork.com/recipes/paula-deen/french-quarter-beignets-recipe-2014039
Some quick online searching yielded that the Cafe du Monde in New Orleans uses cottonseed oil (which has a very high smoke point) and heats their oil to at least 380 degrees. (Source on this,
https://www.pratesiliving.com/the-secrets-behind-cafe-du-mondes-beignets/). Cooking them at a lower temperature, or in not enough oil to maintain a more constant temperature, could definitely effect the texture too.
Hmm, I was looking to see if Paula Deen had a video on this to allow you to see the consistency of the dough, and instead I stumbled on a video from the Cottonseed Oil Comeback Tour folks, then looked up their recipe. Interestingly, they say their recipe is "adapted" from the Paula Deen recipe, but the only adaptation I see is a half-cup less flour, and oil temp of 370 instead of 350. Video:
and recipe
http://www.cottonseedoiltour.com/mardi-gras/ It definitely looks like their hand-mixed dough is still at least a little tacky before the proofing, more like soft dinner rolls than French bread dough.
I've seen recipes with a higher yeast-to-flour ratio. Generally in baking one package of yeast is used to rise 0-4 cups flour, while with seven that's a heavy lift. You're not raising a 6" bread structure but you do want lift and air bubbles. So, if you're experimenting, I'd try increasing the yeast a bit or cutting back at least 1/2 cup of flour, and raise your oil temp.
One other thing that I'm betting a more experienced baker can weigh in on, but I'm guessing that in general you'll get a more flavorful final product in general, and in the end it will tend to level the "how long should I knead" question, if you mix the dough the day before you are going to be making beignets, place into a greased container, cover and refrigerate. It will slow down the yeast action, allowing it to develop more flavor, and as with those no-knead bread recipes, the slower rise also yields a more evenly hydrated dough, with more plentiful air bubbles. If you did that, take from the fridge and allow to come up to room temperature before you move on to rolling out the dough and cutting.
Finally, some other recipes I have seen do a second proof, half an hour to 45 minutes, on the squares after they are cut. That would put some lift back into them, though they may be harder to handle getting into the hot oil. Maybe proof them on strips of parchment, and take the strips right to the oil, gently easing them off?