Substituting fresh shitake mushrooms for dried

Based on my experience with dried and fresh shiitake mushrooms, and the guidance from Gaku Homma in his book Japanese Country Cooking, I'd say that you will simply get different results with fresh shiitake.

There's actually a fair amount of flavor concentration that happens as a side effect of dehydrating mushrooms, and this is particularly pronounced with shiitake (and, for example, porcini). It's not possible to extract these flavors by boiling/simmering fresh shiitake in a way that will match the results of dried. Fresh shiitake will have a superior texture (especially if you are grilling or frying the mushrooms) to the dried ones, but for the purposes of extracting flavor, there is actually a functional advantage to dried ones. This is true for other common Japanese soup stock ingredients, too; nobody I know would use fresh kombu for a soup stock, but they'd consider using it for a salad or salt pickles; dried kombu is more useful for developing flavors in simmered dishes. You'd also get a very different result from fresh skipjack tuna than you'd get from the smoked, fermented, dried katsuobushi.

Any chance you have any other dried mushrooms? I think dried porcini work really nicely for sauces and even Japanese-style soup stock.

In any event, you may get a perfectly serviceable brown sauce with fresh shiitake, but I don't think it will have the same flavor as the version using dried mushrooms. You may be able to simulate drying putting fresh shiitake directly over a gas flame to char, then sauteeing with a flavorful fat. Butter or perhaps a gently heated pumpkin seed oil could work nicely. I don't think there's a simple ratio conversion that you could use for this purpose; I'd use a much larger number of fresh ones than I'd use dried if I wanted to maximize flavor and aroma.