Hi, Double T, welcome to BYC.
There are many brown egg layers. All have easy going personalities. It is the white egg layers that act erratic and struggle, I think they have a stronger flight-or-fight reflex.
Your next request is LARGE eggs and LOTS of them.
This will take you to Delawares, Black Sex Links, Rhode Island Red versions, Barred Rocks or White Rocks, and Australorps.
The Rhode Island Red varieties go by many names: Cherry Eggers, Production Reds, Red Sex Link, Comets, Gold Sex Links, Indian River, ISA Browns, Cinnamon Queens, etc.
Cherry Eggers and Production Reds are pure Rhode Island Red bred for lots of eggs, not appearance. New Hampshires are a takeoff of Reds from early in the century.
The Sex Links, Comets, Indian Rivers, and many other names, are Reds crossed with several other large egg layers: Delawares, White or Barred Rocks
These are top egg layers and the hatcheries have large numbers of them so they are still available. Non-production breeds sell out anytime between January and March determined by demand that year.
Hatcheries that will drop your box of chicks in the mail to you:
Cackle Hatchery
Welp Hatchery
Marti's Poultry
Ideal Hatchery
McMurray
Privett Hatchery
Purely Poultry
Stromberg's
Hoover's Hatchery
There are many more and just google "chicken hatcheries."
If you weaken and want a white egg layer, there are Austra Whites: Australorps crossed with Leghorn, to make a calmer large egg layer that lays lots. Also the Barred Hollands are said to be calm though eggs might not be white white.
About broodies: they decide when to go broody. What you want has no effect - bribes, blackmail, etc nothing works. The hens that lay a lot of eggs have had the broodiness bred out of them, although as individuals, some can still rarely answer the call. It is a hormonal change that triggers the broodiness.
So simply get one or a few of a kind known for her broodiness. I've had great luck with Columbian Wyndottes. Went broody after laying only a month. She'd raise a brood, lay for about a month or six weeks, then go broody again. Great mother - didn't abandon the nest, was smart enough to get food and water and take a break and got back to it. Took great care of her chicks till they feathered out completely. Columbian Wyndottes are known for their broodiness. The broodiness has been deliberately bred out of Silver Lace Wyndottes.
There is a thread on here for Russian Orloffs - who lay great the first year - so you get lots of eggs - then go broody in their second year, to raise up your replacement hens. Great combination I think.
Games and Cochins are broody, too, but don't lay large eggs generally, though some individuals might.
Might be lucky having an Easter Egger going broody.
There are many more great brown egg layers but either the egg size drops to medium, or the number of eggs drops. Speckled Sussex, Dorking, Brahmas, Buckeyes, Wyndottes of several colors, Buff Orpingtons, Marans, Welsummers, Dominiques, Salmon Faverolles, and more rarer ones, etc.
You said Large eggs so I took you to mean really large, like "large" to "extra Large", not medium to medium large; why I mentioned ones I did.
Also, first eggs are smaller, and as the hen lays over the months, the eggs get larger. Two year old hens might lay extra large eggs consistently. Also, amount of good protein affects egg size.
Good Luck. Lots of fun.
Ideal Hatchery sends out a price list with a list of egg size by breed. That is helpful. It is on the web somewhere too.