If you are interested in specific hotspot missense mutations, you may want to query MuPIT (shameless plug) with your gene of interest. It lists amino acid residues that have statistically significant higher local mutation density in 3D protein structures (i.e. missense hotspots) categorized by 31 TCGA cancer types. This is probably a more objective way to indicate whether mutations at a certain residue are cancer drivers rather then eyeballing recurrent codon positions in cBioPortal. From my work developing the statistical algorithm for detecting missense hotspots, there really are not driver missense mutations that appear in absolutely all cancer types, which speaks to the heterogeneity in cancer, although a few well known genes have mutations that appear in the majority of cancer types available in TCGA.
You can follow Cyriac’s suggestion of identifying relevant cancer driver genes. Some of the most prominent for hotspot missense mutations across cancer types which may fit what you are looking for are FBXW7 (residues 465, 479, 505, etc.), PIK3CA (1047, 542, 545, etc.), KRAS (12, 13, and 61), HRAS (12, 13, and 61), BRAF (600, etc.), and TP53 (many residues). You can query your gene by using the following format for the URL: http://mupit.icm.jhu.edu/?gene=GENENAME, for example KRAS would be http://mupit.icm.jhu.edu/?gene=KRAS. Looking at the "TCGA 3D Mutation Hot Regions" column, you can see which cancer types have hotspots. Clicking the "+" button will show you how many hotspot regions there are. Hovering over each hotspot region, for example "hr_1" will pop up a tool tip telling you which residues have statistically significant hotspot mutations. To show the hotspot region on the protein structure, click on the hotspot region (e.g. "hr_1"). One inconvenience, though, is that sometimes protein structures have different residue numbering conventions than typically used by the gene, for example some protein structures of BRAF may list a hotspot at residue 599 rather than 600, but it is actually the same residue. You can switch between protein structures by clicking on a different PDB ID in the left column. The benefit of looking at hotspots on protein structures is that it may give you a clue about the functional effect of the missense mutation. You could then follow Cyriac's suggestion of using cBioPortal for obtaining the mutation data for the relevant hotspot residues.