Last spring I found that I had overdone the over winter mulch, and when I removed it to start planting the soil was still frozen - I think it was about this date, or very close, and outside of the mulched beds the ground had thawed at least two weeks earlier. Ideally I would want some sort of mulch bin/crib/storage right next to the vegetable beds, where I could collect mulch as it was available, and also temporarily dump mulch from the beds during warm/sunny periods early in spring. The snow cover and winter temperatures vary too much here to feasibly mulch the beds thickly enough to not freeze.
Anyway, this year the (mostly leaf) mulch was not that thick, and I have taken most of it off and sowed mixed fava beans a couple of days ago. Yesterday I also planted some potatoes, and as an experiment I put two fava beans near each potato. It will be interesting to see how this works. Although these are supposed to be early potatoes, I think the favas will be harvestable when the potatoes are ready, or maybe a bit earlier. I much prefer green favas to dried, so although I need to have an area planted with favas that can be left until the seed is mature and dry, this only needs to be big enough to provide seed for next year and a little backup. And to ensure a bit of genetic variation is maintained.
Pole beans I may sow in a week or two, but I'm wary of covering the seeded beds with mulch like the OP does - I feel this would invite slugs and other pests at just the most critical time here. A little mulch when they are up and vining, however, has worked well before.
As a side note (not beans), I also sowed some peas along with oats as an experiment. Unlike the favas and potatoes, I won't necessarily eat either, the hope is they will contribute to soil health both as a break in the rotation and by adding organic material, and if enough grows it will make nice snacks for the hens.