I wouldn’t recommend broadcasting seed on top of thick residue.
We have broadcast and stomped he seed in with the cattle but it is a lot of work, takes precise timing, and is risky.
We did the above successfully but the residues were from a standing pear millet and cowpea cover crop on previously bare land and those residues had broken down a lot already when we seeded and stomped so there was a lot of spaces for the seed to fall through
It looks like you have a decent grass stand under that hay but it’s hard to tell from the pic
Is the field resting now or do you plan to have the cattle back on it?
Did your hayfields have any herbicides in the last three years and if so which ones?
Does the hay have visible seeds and if so what kinds?
We have drilled seed successfully over our winter hay feeding areas but we only did this on about a five acre area where we had really destroyed the pasture and left about a foot thick residue of hay, manure and urine as it was an access area to the one watered we had the first winter and we had to keep the cows accessing it
So we kept the mud away by layering the hay and in spring we seeded perennial fescue, ryegrass, clover and some raddish (not normally a spring plant but this was a trial and actually the radish did well…I had been concerned with the urine and it being a harsh environment to spend a lot of money on expensive seed)
– we used a great seed inoculant that had minerals and microbes and enzymes to get it going fast and give it starting nutrient
So if you can rest that area long enough for the cover crop to emerge, and you look at the costs and want to spend the time and money, and you know how to calibrate the drill and get the seed depth and mixtures right, then you could easily drill it.
If you have animals without chemical wormers, don’t use herbicides, insecticides or other chemicals, hay without herbicides, got a pretty good cover on there, and have seeds in the hay, you shouldn’t need to seed it to get good spring germination of perennials.
You will have to block it off until at least late April or possibly longer, until those plants have 4 leaves.
If you don’t want perennials, or have destroyed it a lot, you could drill.
What you plant depends on your end goal.
Oats and clover, or oats and and peas, followed by tillage raddish with crimson clover rye and/ or ryegrass could work.