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Dissolve the yeast and sugar in warm water. Let bloom for 10 minutes.
Add 3 C flour, 2 room temperature eggs, salt, and buter. Beat with a wooden spoon until you form a smooth dough that pulls away from the bowl. It should come together very quickly. Turn out the dough onto a counter and knead in the remaining 1/2 cup flour until smooth and shiny, about five minutes. Put the dough into an oiled bowl and let rise for about one hour. Punch down the dough and let it rise again if you have (or need) the time; I let mine rise for another 30 minutes.
Pull and cajole the dough into a large circle and lay it in a tart pan or pie tin - with at least a 1-2" overhang. Roll your fruit in sugar then spoon it into the crust. Dot it with butter. Fold the dough inward to create a free-form galette. Brush the surface with a beaten egg and sprinkle with a little bit more sugar.