Well, it's definitely doable to create a flock of chickens that looks like Egyptian Fayoumis with both upped egg production and increased broodiness with the chickens that you have, but it will likely take a few years. I'll go over two or three different ways that you could do this: the long and careful way, or some much easier but less precise ways. In each way, I suggest completely replacing your flock with the product for the best results.
Long an careful: You should breed your Leghorn roosters to your Egyptian Fayoumi hens and putting their eggs under your broodies. From there, you would select the offspring with the most Fayoumi-like features and with the best egg production, then breed some of those mixed hens back to Fayoumi roosters, and the rest of them back to their most Fayoumi-like brother and put their eggs under your broodies. From there, you should select the offspring with the best traits in each group and breed your best hens to your OEG roosters and let your broody hens hatch the eggs. Then again, select the offspring with the most desirable traits, and put them all together and let them naturally breed and raise chicks. Select towards the chickens with the best traits in each generation, and after two or three generations you should be good. This would make a whole new flock and take probably around five years.
Easier way: Breed some Egyptian Fayoumi hens to OEG roosters, and some to Leghorn roosters. Select for the best offspring from each group and breed them with eachother (roos from group A with the hens from group B, males from group B with the hens from group A.) Select for the most desirable offspring, and either 1. (easiest) cultivate a flock from them, selecting for the most desirable ones in each generation, or 2. (would take a little more patience) breed them back to Egyptian Fayoumis. Select the most desirable offspring and cultivate a flock from them, selecting the best ones each generation. This would take an extra year of selective breeding, but would result in less culls in the next few generations.
The basic principle is just mix them together and select. And of course, if you prefer a less hands-on approach, you can just add some OEGB and Leghorns to your flock and let them integrate over the next few generations. But of course, that would result in a much less precise result.