That recipe is so far from what most cooks would consider a pizza recipe1, that I am ignoring the unusual ingredients for now.
What is blatantly obvious is the hydration2 - 250 ml of milk plus an egg for 1.5 kg of flour means around 20%. This won’t give you a workable yeast dough. The standard pizza recipe is somewhere between 60 and 80%, depending on the kind of pizza crust and the types of flour used. This website gives a nice comparison of the different types.
If you want to salvage the current batch, you will have to gradually incorporate around 500-700 ml of water and yes, this will initially get you to a “solid bits in muddy liquid” stage. It would be easiest if you could dump it in a stand mixer. First it will be a lump sloshing around in water. After a while, the lumps should gradually soften and the dough becomes even. Work in batches, if necessary, not all mixers can deal with this much dough. If you are working by hand, use a large bowl for the initial “mushing” and later you can go back to the bench.
General advice: Don’t let “sticky” dough tempt you to constantly flour the bench, you’d be throwing the ratios again. Rather learn about different kneading techniques, some are great for soft dough.
1 A standard pizza recipe is “flour, water, salt, yeast and (optional) oil”. Use good quality ingredients, very little yeast and lots of time. Your is some kind of enriched dough, which I would expect for other things like challah, sweet buns, ...
2 Hydration or Baker’s Percentage is the percentage of liquid with the flour as base unit. Your average bread dough will be around 60% and 65%, bagels can be as low as 50% and going over 70% brings you into high-hydration territory where regular kneading won’t work too well. Find an overview here.